Notes in LinAl_SuperDecks

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Status	Last Update	Fields
Published	09/22/2024	Toeplitz: elements are {{c1::constant}} along {{c2::each diagonal}}.
Published	09/27/2024	Matrix \(A\) is Toeplitz if its elements \(a_{ij}\) have the form \(a_{ij}\)={{c1::\(f(i-j)\)}}
Published	09/22/2024	Fact 2.2. A \(N \times N\) matrix \(A\) is circulant iff its elements \(a_{ij}\) have the form \(a_{ij} = {{c1::f(\text{mod}(i - j, N))}}\) for some f…
Published	09/23/2024	Q2.2: True or False: All circulant matrices are Toeplizt
Published	09/23/2024	Def: Matrix \(A\) is called symmetric when? Hint: write concisely in math notation
Published	09/23/2024	Def: Matrix \(A\) is called Hermitian symmetric when? Hint: write concisely in math notation
Published	09/23/2024	Properties of Hermitian Transpose: Involution
Published	09/23/2024	Properties of Hermitian Transpose: Scaling
Published	09/23/2024	Properties of Hermitian Transpose: Multiplication \[(AB)'\ =\ ?\]
Published	09/23/2024	(2.5) Inner product: Write the inner product as a summation, using the hint belows:\[\langle \mathbf{x}, \mathbf{y} \rangle = \mathbf{y}^\prime \…
Published	10/20/2024	(2.5) Outer product: Think about what the final matrix of the outer product will look like using the following hint:\[\mathbf{x} \mathbf{y}^\prime = …
Published	09/23/2024	(2.5) Unit vector: What would \(\mathbf{e}_2\) looks like in \(\mathbb{R}^4 \,\)?
Published	09/23/2024	~~SAMPLE BASIC
Published	09/22/2024	~~SAMPLE CLOZE {{c1::samplecloze}}
Published	09/23/2024	How many eigenvalues a square matrix \( \mathbb{A} \in \mathbb{F}^{N \times N} \) has?
Published	09/23/2024	Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a singular matrix, what can we say about vector \( \mathbf{v} \) in the follo…
Published	09/23/2024	Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a nonsingular matrix, what can we say about vector \( \mathbf{v} \) in the fo…
Published	09/23/2024	Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a singular matrix, is the determinant of  \(\mathbf{A}\) equal to 0…
Published	09/23/2024	Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a singular matrix, is \(\mathbf{A}\) invertible?
Published	09/23/2024	Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a non-singular matrix, is the determinant of  \(\mathbf{A}\) equal …
Published	09/23/2024	Singular Matrix: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is a non-singular matrix, is \(\mathbf{A}\) invertible?
Published	09/23/2024	Eigenvectors: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) has \( N \) eigenvalues\( \lambda_1, \dots, \lambda_N \in \mathbb{C} \), …
Published	09/23/2024	Eigenvectors: If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) has \( N \) eigenvalues\( \lambda_1, \dots, \lambda_N \in \mathbb{C} \). …
Published	09/23/2024	Eigenvectors: For an \( N \times N \) matrix \( \mathbf{A} \), let \( \lambda_1, \dots, \lambda_N \) denote its \( N \) eigenvalues. For each eig…
Published	09/23/2024	Eigenvectors: For an \( N \times N \) matrix \( \mathbf{A} \), let \( \lambda_1, \dots, \lambda_N \) denote its \( N \) eigenvalues. For each eig…
Published	09/22/2024	Spectral Theorem Special Case:If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is (Hermitian) symmetric, i.e., \( \mathbf{A} = \mathbf{A}' \), then the…
Published	09/22/2024	Spectral Theorem Special Case:If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is (Hermitian) symmetric, i.e., \( \mathbf{A} = \mathbf{A}' \), then the…
Published	09/23/2024	Spectral Theorem Special Case:Given:\( \mathbf{A} \in \mathbb{F}^{N \times N} \) is (Hermitian) symmetric, i.e., \( \mathbf{A} = \mathbf{A}' \)\(\math…
Published	09/23/2024	Spectral Theorem Special Case:\( \mathbf{A} \in \mathbb{F}^{N \times N} \) is (Hermitian) symmetric, i.e., \( \mathbf{A} = \mathbf{A}' \)\(\mathbf{V} …
Published	09/23/2024	Spectral Theorem deals with the ability to diagonalize a normal matrix. What is/are the special case that spectral theorem applies to?
Published	09/23/2024	Spectral Theorem deals with the ability to diagonalize a normal matrix. The special cases that spectral theorem applies to involve a matrix \(\mathbf{…
Published	09/22/2024	Normal matrices:A matrix \( \mathbf{A}\) is normal if and only if {{c1::\( \mathbf{A}' \mathbf{A} = \mathbf{A} \mathbf{A}' \)}}
Published	09/22/2024	Fact 3.2. The spectral theorem says: A square matrix \( \mathbf{A} \) is diagonalizable by a unitary matrix if and only if \( \mathbf{A} \) …
Published	09/23/2024	True or False: Permutaion matrix is a normal matrix.
Published	09/23/2024	True or False: Permutaion matrix is a orthogonal matrix.
Published	09/23/2024	True or False: Permutaion matrix properties \( \mathbf{P}^{-1} = \mathbf{P}' \)
Published	09/23/2024	True or False: orthogonal matrix is normal
Published	09/23/2024	True or False: Permutaion matrix has a unitary eigendecomposition.
Published	09/23/2024	True or False: If \(\mathbf{A}\) and \(\mathbf{B}\) are normal matrices, and thus have eigendecompositions guaranteed by Spectral Theor…
Published	09/23/2024	True or False: If \(\mathbf{A}\) and \(\mathbf{B}\) are normal matrices, and thus have eigendecompositions guaranteed by Spectral Theor…
Published	09/23/2024	What does it mean to say that a square  matrix \(\mathbf{A}\) is similar to a diagonal matrix?
Published	09/23/2024	True of False:If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) has \( N \) linearly independent eigenvectors \( \mathbf{V} = [\mathbf{v}_1 \dots \mathb…
Published	09/23/2024	True of False:  A diagonalizable matrix is also invertible.
Published	09/23/2024	A matrix is non-invertible (singular) if and only if its columns (or rows) are {{c1::Linearly Dependent}}
Published	09/23/2024	True of False:If \( \mathbf{A} \in \mathbb{F}^{N \times N} \) has \( N \) linearly dependent eigenvectors \( \mathbf{V} = [\mathbf{v}_1 \dots \mathbf{…
Published	09/23/2024	Fact 3.4. Every matrix \( A \in \mathbb{F}^{N \times N} \) is similar to a matrix in Jordan normal form:\[A = P J P^{-1}\]for an invertible matrix \( …
Published	09/22/2024	Let \( A \in \mathbb{F}^{N \times N} \) be a (Hermitian) symmetric matrix, so \( A = V \Lambda V' \).Consider the linear transform \( x \mapsto y = A …
Published	09/22/2024	Let \( A \in \mathbb{F}^{N \times N} \) be a (Hermitian) symmetric matrix, so \( A = V \Lambda V' \).Consider the linear transform \( x \mapsto y = A …
Published	09/22/2024	Let \( A \in \mathbb{F}^{N \times N} \) be a (Hermitian) symmetric matrix, so \( A = V \Lambda V' \).Consider the linear transform \( x \mapsto y = A …
Published	09/23/2024	True or False: Given a square diagonalizable matrix \( A \in \mathbb{F}^{N \times N} \).\[ A = V \Lambda V^{-1} \implies A^k = V \Lambda^k V^{-1}…
Published	09/23/2024	True or False: If \( A \) is diagonalizable, then \( A^k \) is diagonalizable for all \( k \in \mathbb{N} \)?
Published	09/23/2024	True or False: If \( A \) is invertible, then \( A^k \) is invertible for all \( k \in \mathbb{N} \)?
Published	09/23/2024	A matrix \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is \textbf{invertible} if and only if the following statement holds: • the eigenvalues of \…
Published	09/23/2024	Odie
Published	09/23/2024	Fact 3.5. Every \( V \in \mathbb{R}^{2 \times 2} \) with \( V' V = I_2 \) (i.e., orthogonal) has the following form for some rotation angle \( \theta …
Published	09/23/2024	Q.3.3 True or False: Every permutation matrix has one real eigenvalues
Published	09/23/2024	Q.3.4 True or False: There exists a permutation matrix with all real eigenvalues
Published	09/23/2024	Q.3.1 True or False: Every unitary matrix is a normal matrix
Published	09/23/2024	Q.3.1 True or False: Every normal matrix is a unitary matrix
Published	09/23/2024	Q3.5 If \( A \in \mathbb{R}^{2 \times 2} \) has real eigenvalues \( \lambda_1, \lambda_2 \), then the locus of points \[\{ Ax : x \in \mathbb{R}^2 \te…
Published	09/23/2024	Define. A {{c1::non-negative real number}} \( \sigma \) is called a singular value of a matrix \( A \in \mathbb{F}^{M \times N} \) if there exist {{c2…
Published	09/23/2024	Fact 3.6. A \( M \times N \) matrix has at most {{c1::\( \min(M, N) \)}} distinct singular values.
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that\[X = U \Sigma V' \]What is …
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that:\[X = U \Sigma V' = \sum_{k…
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that:\[X = U \Sigma V' = \sum_{k…
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that:\[X = U \Sigma V' = \sum_{k…
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that:\[X = U \Sigma V' = \sum_{k…
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that:\[X = U \Sigma V' = \sum_{k…
Published	09/23/2024	Fact 3.7. Existence of SVD: If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) such that:\[X = U \Sigma V'\]This fac…
Published	09/23/2024	True or False: The singular values in a SVD can be complex.
Published	09/23/2024	If \( X \in \mathbb{F}^{M \times N} \), then there exist matrices \( U, V, \Sigma \) for a SVD such that:\[X = U \Sigma V' = \sum_{k=1}^{\mi…
Published	09/23/2024	Given an SVD where \( A = U \Sigma V' \).If \( v_i \) is the \( i \)-th column of \( V \), and \( u_i \) is the \( i \)-th column of \( U \), the…
Published	09/23/2024	Given an SVD where \( A = U \Sigma V' \).If \( v_i \) is the \( i \)-th column of \( V \), and \( u_i \) is the \( i \)-th column of \( U \), the…
Published	09/23/2024	Given an SVD where \( A = U \Sigma V' \).Scaling property: if \( B = \alpha A \), then \( B = \) {{c1::\(U \alpha\Sigma V' \)}}
Published	09/23/2024	Given an SVD where \( A = U \Sigma V' \).If \( B = A Q' \) where \( Q \) is a unitary matrix, then an SVD of \( B \) is \( B = \){{c1::\(U \Sigma (QV)…
Published	09/23/2024	What is the solution \(x_{\star}\) to the optimization problem described by:\[x_{\star} = \arg \max_{x \in \mathbb{F}^N : \\| x \\|_2 = 1} \\| \math…
Published	09/23/2024	tester AnkiCollab
Published	09/23/2024	Complete the definition of matrix 2-norm (also known as spectral norm or operator norm):\[\|\Vert A \|\Vert_2 \triangleq \max_{x: \Vert x \Vert_2 =…
Published	09/23/2024	What is value that the definition of matrix 2-norm (also known as spectral norm or operator norm) evaluated to?\[\|\Vert A \|\Vert_2 \triangleq \ma…
Published	09/23/2024	Spectral norm of matrix \(\mathbf{A}\) is denoted as \(\|\Vert A \|\Vert_2\). The spectral norm gives a tight upper bound on how much the Euclidean norm…
Published	09/23/2024	Spectral norm of matrix \(\mathbf{A}\) is denoted as \(\|\Vert A \|\Vert_2\). The spectral norm gives a tight upper bound on how much the Euclidean norm…
Published	09/23/2024	This is not a questionFact 3.9. The solution \( x_{\star} = v_1 \) is unique to within a phase factor \( e^{i\phi} \) iff \( \sigma_1 > \sigma_2 \)…
Published	09/23/2024	Smallest singular value:Given an \( M \times N \) matrix \(A\) with \(N \leq M\), the smallest singular value of \(A\) also corresponds to an opt…
Published	09/23/2024	Fact 3.10. If \( A \) is an \( N \times N \) Hermitian matrix, with eigenvalues ordered as \( \lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_N \), t…
Published	09/23/2024	Fact 3.10. If \( A \) is an \( N \times N \) Hermitian matrix, with eigenvalues ordered as \( \lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_N \), t…
Published	09/23/2024	Fact 3.10. If \( A \) is an \( N \times N \) Hermitian matrix, with eigenvalues ordered as \( \lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_N \), t…
Published	09/23/2024	Fact 3.10. If \( A \) is an \( N \times N \) Hermitian matrix, with eigenvalues ordered as \( \lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_N \), t…
Published	09/23/2024	Any {{c1::right singular vector}} of \({{c2:: A }}\) is an {{c4::eigenvector}} of \({{c3:: A'A }}\).
Published	09/23/2024	Any {{c1::left singular vector}} of \({{c2:: A }}\) is an {{c4::eigenvector}} of \({{c3:: AA' }}\).
Published	09/24/2024	Relating SVDs and Eigendecompositions:The {{c1::singular values}} of a matrix \(\mathbf{A} \in \mathbb{F}^{M \times N}\) are the {{c2::sorted squ…
Published	09/24/2024	If \( A \) is a normal \( N \times N \) matrix (e.g., Hermitian symmetric), then it has a unitary eigendecomposition. The following are the steps towa…
Published	09/24/2024	I hate this question. Flip if you must. Fact 3.12. If \( A \) is normal and has eigenvalues with distinct magnitudes, or if eigenvalues with equa…
Published	09/24/2024	I hate this question. Flip if you must. Fact 3.12. Every unit-norm eigenvector of normal \(A \) is also a {{c1::right singular vectors}…
Published	09/24/2024	Fact 3.13. An SVD of \( A \) can have \( U = V \) if and only if \( A \) is {{c1::Hermitian symmetric, \( A = A' \)}}, and all eigenvalues of \( A \) …
Published	09/24/2024	Q3.9 True or False: \text{Q3.9 If } \[A = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 2 \end{bmatrix}, \text{ …
Published	09/24/2024	Q3.10 True or False: The singular values of a permutation matrix all equal to 1
Published	09/24/2024	Q3.11 True or False: (A uniqueness question.) If  \[A = U \Sigma V' \text{ and } A = \tilde{U} \tilde{\Sigma} \tilde{V}' \text{ are bot…
Published	09/24/2024	By definition, to determine if \( x \) is an eigenvector of a square matrix \( A \), simply compute \( y = A x \) and see if \( y = \alpha x \) for so…
Published	09/24/2024	For an \( M \times N \) matrix \( A \), how do we test if \( v \in \mathbb{F}^N \) is a right singular vector or \( u \in \mathbb{F}^M \) is a left si…
Published	09/24/2024	Fact 3.14. If \( A = U \Sigma V' \) has {{c1::distinct}} singular values, then every {{c2::right singular vector}} of \( A \) is a multiple of some co…
Published	09/24/2024	Define. A \( N \times N \) (square) Hermitian matrix \( A \) is {{c1::positive semidefinite}} iff {{c2::\[ x' A x \geq 0 \quad \text{for all} \quad x …
Published	09/24/2024	Define. A (square) Hermitian matrix \( A \) is {{c1::positive definite}} iff\[{{c2::x' A x > 0 \quad \text{for all} \quad x \neq 0.}}\]
Published	09/24/2024	Fact 3.15. If \( A = BB' \) for any matrix \( B \), then \( A \) is {{c1::positive semidefinite.}}
Published	09/24/2024	Q3.14 Which of the following statements is true?A: All positive-semidefinite matrices are positive definite.B: All positive-definite matrices are posi…
Published	09/24/2024	Fact 3.16. If \( A = BB' \) for any matrix \( B \), then eigendecomposition of \( A \) is also an {{c1::SVD}}
Published	09/24/2024	Q3.15 If \( A \) is Hermitian, then \( A = BB' \) for some matrix \( B \).A: TrueB: False
Published	09/24/2024	Fact 3.17. A Hermitian matrix \( A \) is {{c1::positive semidefinite}} if and only if all of its eigenvalues are {{c2::nonnegative}}.
Published	09/24/2024	Fact 3.17. A Hermitian matrix \( A \) is {{c1::positive definite}} if and only if all of its eigenvalues are {{c2::positive}}.
Published	09/24/2024	Fact 3.18. A Hermitian matrix \( A \) is {{c1::positive semidefinite}} if and only if we can write {{c2::\( A = BB' \) for some matrix \( B \)}}.
Published	09/24/2024	Define. For a vector space \( \mathcal{V} \) defined on a field \( \mathbb{F} \), a nonempty subset \( S \subseteq \mathcal{V} \) is called a subspace…
Published	09/24/2024	Q4.1 True or False: Is the subset of orthogonal matrices \( S = \{ A \in \mathbb{R}^{N \times N} : A' A = I \} \)a subspace of \( \mathbb{R}^{N \…
Published	09/24/2024	Define. Given a set of vectors \( \{ \mathbf{u}_1, \dots, \mathbf{u}_N \} \) in a vector space \( \mathcal{V} \) over field \( \mathbb{F} \), the \tex…
Published	09/24/2024	Fact 4.2. If \( \mathbf{u}_1, \dots, \mathbf{u}_N \in \mathcal{V} \), then the \( \text{span}(\{ \mathbf{u}_1, \dots, \mathbf{u}_N \}) \) is a {{…
Published	09/24/2024	Define. In a vector space \( \mathcal{V} \), what is the span of the empty set \( \emptyset \) equal to?\[\text{span}(\emptyset) \triangleq\ ?\]
Published	09/27/2024	Define. A set of vectors \( u_1, \ldots, u_N \) in a vector space is linearly independent iff, for any scalars \( \alpha_1, \ldots, \alpha_N \in \math…
Published	09/27/2024	If a set of vectors \( u_1, \ldots, u_N \) is linearly dependent what can be said about the tuple of coefficients \( \alpha_1, \ldots, \alpha_N \…
Published	09/27/2024	Any set of vectors containing a zero vector is a linearly {{c1::dependent}} set.
Published	09/27/2024	A set consisting of a single nonzero vector is a linearly {{c1::independent}} set.
Published	09/27/2024	Any set of orthonormal vectros is a linearly {{c1::independent}} set
Published	09/27/2024	Generalizing Linear Indepence into Infinite SetDefine. A (possibly uncountably infinite) set of vectors \(\mathcal{X}\) in a vector space is linearly …
Published	09/27/2024	Fact 4.3. A set of vectors \(u_1, \ldots, u_N \in \mathcal{V}\) is linearly independent iff the \(N \times N\) Gram matrix \(G\) is {{c1::invertible}}…
Published	09/27/2024	Define. A set of vectors \(\{ b_1, b_2, \ldots \}\) in a vector space \(\mathcal{V}\) is a basis for a subspace \( \mathcal{S} \subseteq \mathcal{V} \…
Published	09/27/2024	Can \(\mathbf{0}\) vector be in a basis?
Published	09/27/2024	True of False: every subspace in a vector space has a basis. (FYI question, not too important)
Published	09/27/2024	Fact 4.7. If \(\{ b_1, \ldots, b_N \}\) is a basis for \(\mathcal{S} \subseteq \mathcal{V}\) for \(N \in \mathbb{N}\), then every vector \( v \in \mat…
Published	09/29/2024	Fact 4.7. If \(\{ b_1, \ldots, b_N \}\) is a basis for \(\mathcal{S} \subseteq \mathcal{V}\) for \(N \in \mathbb{N}\), then every vector \( v \in \mat…
Published	09/29/2024	Fact 4.8. Every basis for a subspace has the same {{c1::cardinality::the word for number of elements}} (possibly infinite).
Published	09/29/2024	Q4.3 The vector space of sinusoids of frequency \(\nu\) is: \(\mathcal{V} = \{ A \cos(2\pi\nu t + \phi), \, t \in \mathbb{R} \, : \, A, \phi \in \math…
Published	09/29/2024	Q4.4 Let \(\mathcal{V} = \mathbb{R}^3\) and \(\mathcal{S} \triangleq \{(1,1,1), (1,0,0)\}\). What is \(\text{span}(\mathcal{S})\)?A: 0 B: a lineC: a p…
Published	09/29/2024	Q4.5 The set of \(N \times N\) upper triangular matrices is a subspace of \(\mathbb{F}^{N \times N}\).A: True B: False
Published	09/29/2024	Q4.6 A basis for such upper triangular matrices in vector space \(\mathbb{F}^{N \times N}\) contains how many vectors?A: \(N\) B: \(N^2\) C:…
Published	09/29/2024	Define. The {{c1::dimension}} of a subspace \(\mathcal{S}\) is the number of elements in any basis for \(\mathcal{S}\)
Published	09/29/2024	Q4.7 What is the dimension of the subspace in \(\mathbb{F}^{N \times N}\) of \(N \times N\) diagonal matrices? A: 1 B: \(N\)C: 2\(N\)D: \(N^2\)E: \(\i…
Published	09/29/2024	Q4.8 What is the dimension of the vector space pf polynomial with even powers?A: 1 B: 2C: infiniteD: undefined
Published	09/29/2024	Not a questionDefine. There are two types of subspaces (and vector spaces).A finite-dimensional (sub)space has a basis with \(\dim \in \{ 0 \} \cup \m…
Published	09/29/2024	Fact 4.9. If \( S = \text{span}( \{ u_1, \dots, u_N \} ) \), where each \( u \in \mathcal{V} \), then \( \dim(S) \leq {{c1::\min(N, \dim(\mathcal{V}))…
Published	09/29/2024	Fact 4.10. The basis reduction theorem states: To form a basis for \(S\):Either \(\{ u_1, \dots, u_N \}\) is a {{c1::basis}} for \(S = \text{span…
Published	09/29/2024	Fact 4.11. The basis extension theorem [3, Thm. 5.37] states that every linearly independent set of \(N\) vectors in a finite-dimensional subspace \(S…
Published	09/29/2024	Fact 4.12. If \( S \) and \( \mathcal{T} \) are both subspaces of a finite-dimensional vector space \( \mathcal{V} \), then\[S {{c2::\subseteq}} \math…
Published	09/29/2024	Define. If \( S, \mathcal{T} \subseteq \mathcal{V} \) then- the sum (or Minkowski sum) of two subspaces is defined as\[S + \mathcal{T} = \{ {{c1::s + …
Published	09/29/2024	Fact 4.13. If \( S, \mathcal{T} \subseteq \mathcal{V} \), then \( S {{c1::+}} \mathcal{T} \) and \( S {{c1::\cap}} \mathcal{T} \) are both {{c2::subsp…
Published	09/29/2024	Q4.9 Considering the same \( S \) and \( \mathcal{T} \), what is \( S \cap \mathcal{T} \)?A: \(\emptyset\) B: diagonal matrices C: tridiagon…
Published	09/29/2024	Are these two expressions the same: \( S + \mathcal{T} \) and union of subspaces \( S \cup \mathcal{T} \)?
Published	09/29/2024	Q.4.10 A union of subspaces is a subspace.A: TrueB: False
Published	09/29/2024	Define. For subspaces \( S \) and \( \mathcal{T} \) of vector space \( \mathcal{V} \), we write the subspace sum \( S + \mathcal{T} \) as a {{c1::dire…
Published	09/29/2024	Q4.11 For \( s, t \in \mathcal{V} \) with \( s, t \neq 0 \) and \( t \neq \alpha s \) for all \( \alpha \), let \( \mathcal{S} = \text{span}(s) \) and…
Published	09/29/2024	Fact 4.14. [1, Theorem 2.26] If \(\mathcal{U} = \mathcal{S} \oplus \mathcal{T}\) then:Every \( u \in \mathcal{U} \) can be written uniquely in the for…
Published	09/29/2024	Fact 4.14. [1, Theorem 2.26] If \(\mathcal{U} = \mathcal{S} \oplus \mathcal{T}\) then:Every \( u \in \mathcal{U} \) can be written uniquely in the for…
Published	09/29/2024	Define. For a subspace \(\mathcal{S}\) of a vector space \(\mathcal{V}\), the orthogonal complement of \(\mathcal{S}\) is the subset of vectors in \(\…
Published	09/30/2024	Q4.12 In \(\mathbb{R}^3\), if \(\mathcal{S}\) is a line through the origin, then what geometric shape is \(\mathcal{S}^{\perp}\)?A: empty setB: pointC…
Published	09/30/2024	Fact 4.15. Key properties of orthogonal complements when \(\mathcal{V}\) is finite dimensional (like \(\mathbb{F}^N\)) [1, Theorem 3.11]:\(\mathcal{S}…
Published	09/30/2024	Fact 4.16. Decomposition theorem for subspaces [1, p. 22]. If \(\mathcal{S}\) is a subspace in \(\mathcal{V}\), then, because \(\mathcal{S} \oplus \ma…
Published	09/30/2024	Define. Let \(\mathcal{V}\) and \(\mathcal{W}\) be vector spaces on a common field \(\mathbb{F}\).  A function \( L : \mathcal{V} \rightarrow \ma…
Published	09/30/2024	The range of a matrix in \(\mathbb{F}^{M \times N}\) is a {{c2::subspace}} of \(\mathbb{F}^{{{c1::M}}}\).
Published	10/01/2024	Define. The row space of a matrix \( A \) is the span of its rows: \[ \mathcal{R}({{c1::A'}}) \].
Published	10/01/2024	Q4.13 If \( D \) is a diagonal matrix in \(\mathbb{R}^{N \times N}\), what is \( R(D) \)?A: \(\emptyset\)B: Usually \(\mathbb{R}^N\)C: Always \(\mathb…
Published	10/01/2024	Fact 4.17. If \( A \) is any \( M \times N \) matrix and \( B \) is an \( N \times K \) matrix with linearly independent rows, then\[R(AB) = R({{c1::A…
Published	10/01/2024	Define. For any \( M \times N \) matrix \( A \):The column rank of \( A \triangleq \mathrm{Dim}({{c1::R(A)}}) = \text{the number of} \text{ {{c1::of l…
Published	10/01/2024	Fact 4.18. For any \( M \times N \) matrix \( A \), its row rank {{c1::equals}} its column rank.
Published	10/01/2024	\[A \in \mathbb{F}^{M \times N} \implies {{c1::0}} \leq \mathrm{rank}(A) \leq {{c1::\min(M, N)}}\]
Published	10/01/2024	Define. When \( A \in \mathbb{F}^{M \times N} \) and \( \mathrm{rank}(A) = \min(M, N) \), we say \( A \) has {{c1::full rank}}.
Published	10/01/2024	Q4.14 What is the rank of  \[\begin{bmatrix} 5 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 7 \end{bmatrix}\]A: 0B: 1C: 2D: 3E: 4
Published	10/01/2024	Fact 4.20. Multiplying matrices never increases rank:  \( A \in \mathbb{F}^{M \times N} \), \( B \in \mathbb{F}^{N \times K} \implies \)\[0 \leq …
Published	10/01/2024	Caution: in general, \(\mathrm{rank}(AB) {{c1::\neq}} \mathrm{rank}(BA)\), even if the sizes are compatible.
Published	10/01/2024	If \( B \) has full row rank (linearly independent rows), then it follows from (4.22) that \(\mathrm{rank}(AB) = \mathrm{rank}({{c1::A}})\).
Published	10/01/2024	Q4.15 If \( A \) has linearly independent columns, then \[\mathrm{rank}(AB) = \mathrm{rank}(A)\]True of False?
Published	10/01/2024	Q4.15 If \( A \) has linearly independent columns, then \[ \mathrm{rank}(AB) = \mathrm{rank}(B)\]True of False?
Published	10/01/2024	Q4.16 For \( u \in \mathbb{C}^{M} \), \( v \in \mathbb{C}^{N} \), what is the {minimum, maximum} possible rank of the outer product \( uv' \)?A: 0, 1B…
Published	10/07/2024	(NOT A QUESTION. I dont think these formulas are important)A lower bound for the rank of a matrix product (Sylvester's rank inequality):\[A \in \mathb…
Published	10/07/2024	Q4.17 If \( A_1, A_2, \ldots, A_K \) are sized appropriately to allow multiplication, then\[\mathrm{rank}(A_1 A_2 \cdots A_K) \leq \min(\mathrm{rank}(…
Published	10/07/2024	Q4.18 For \( u_1, u_2 \in \mathbb{C}^{M} \) and \( v_1, v_2 \in \mathbb{C}^{N} \), what is the {minimum, maximum} possible rank of the \( M \times N \…
Published	10/07/2024	Q4.19 If \( \mathcal{V} = \mathbb{F}^6 \) and \( x \) and \( y \) are two linearly independent vectors in \( \mathcal{V} \), then what is the dimensio…
Published	10/07/2024	Fact 4.21.\[Q \in \mathbb{F}^{M \times M} \, \text{and} \, Q \, \text{unitary} \implies \text{rank} \left( Q A \right) = {{c1::\text{rank} \left( A \r…
Published	10/07/2024	Fact 4.22. (A corollary of fact 4.21) if \( A \) is Hermitian (or normal) with unitary eigendecomposition \( A = V \Lambda V' \) then, because \(…
Published	10/07/2024	Q4.20 If \( A \) is square, then \(\text{rank}(A) = \#\) of nonzero eigenvalues of \( A \).A: True B: False
Published	10/07/2024	By unitary invariance, if \( A \) has SVD \( A = U \Sigma V' \), then \(\text{rank}(A) = {{c1::\text{rank}(\Sigma)}}\).\(\text{rank}(\Sigma) = r = {{c…
Published	10/07/2024	The SVD expression for a matrix \( A \) having rank \( r \) where \( r \leq \min(M, N) \) can be simplified:\[A \in \mathbb{F}^{M \times N} \impl…
Published	10/07/2024	Define. The null space or kernel of an \( M \times N \) matrix \( A \) is\[\mathcal{N}(A) = \ker(A) \triangleq \left\{ x \in \mathbb{F}^N : {{c1::A x …
Published	10/07/2024	Bin's True or False: We always have \( 0 \in \mathcal{N}(A) \).
Published	10/07/2024	Bin's True or False:\(\mathcal{N}(A)\) is a subspace
Published	10/16/2024	Q4.21 If \( A \in \mathbb{C}^{M \times N} \), then \( \mathcal{N}(A) \) is a subspace of what vector space?A: \( \mathbb{C}^M \) B: \( \mathbb{C}…
Published	10/16/2024	Properties of Null Space:\(\mathcal{N}(A) = {{c2::\{0\} }} \iff A\) has {{c1::linearly independent}} columns
Published	10/16/2024	Properties of Null Space:\(\mathcal{N}(A) = {{c1::\mathbb{F}^N}} \iff A = {{c2::0_{M \times N}}}\)
Published	10/16/2024	Properties of Null Space:\(\mathcal{N}(B) {{c1::\subseteq}} \mathcal{N}(AB)\)
Published	10/16/2024	Properties of Null Space:If \(\mathcal{N}(A) = {{c1::\{0\} }}\), i.e., if \(A\) {{c1::has full column rank,}} then \(\mathcal{N}(B) = \mathcal{N}(AB)\…
Published	10/16/2024	Decomposition theorem for matricesFact 4.25. If \( A \in \mathbb{F}^{M \times N} \), then, from (4.15), the input and output spaces of \( A \) satisfy…
Published	10/16/2024	Decomposition theorem for matricesRecall that every vector \( y \in \mathbb{F}^M \) can be decomposed uniquely as \( y = y_1 + y_0 \), where \( y_1 \i…
Published	10/16/2024	Relationships between null space and range of a matrixFact 4.26. [1, Theorem 3.12]. For any matrix \( A \), its null space and range are related by:\[…
Published	10/16/2024	If \( A \in \mathbb{F}^{M \times N} \) then\[\text{Dim}({{c1::\text{N}(\text{A})}}) + \text{rank}(\text{A}) = N\]
Published	10/16/2024	Q4.22 If \( u \in \mathbb{C}^{M} \setminus \{0\} \) and \( v \in \mathbb{C}^{N} \setminus \{0\} \), what is the nullity of their outer product, i.e., …
Published	10/17/2024	Find dimension of partition matrices:When \( A \in \mathbb{F}^{M \times N} \) has rank \( 0 < r < \min(M, N) \), we can partition the SVD compon…
Published	10/17/2024	Q4.24 If \( x = V_0 z \) for some \( z \in \mathbb{F}^{N-r} \), then what is \( A x \)?A: \( 0 \) B: \( U_r \Sigma_r z \) C: \( U_r V_r z \)…
Published	10/17/2024	Fill in the partition for SVD of a tall matrixWhen \( A \) is tall or "thin," i.e., \( M > N \implies r \leq N < M \), then we can simplify:\[A …
Published	10/17/2024	Fill in the partition for SVD of a wide matrixWhen \( A \) is wide, i.e., \( M < N \implies r \leq M < N \), then we can simplify:\[A = U \Sigma…
Published	10/17/2024	Fact 4.30. The \( N \) columns of any orthogonal matrix \( V \in \mathbb{R}^{N \times N} \) are an {{c1::orthonormal}} basis for \( \mathbb{R}^N \).Fa…
Published	10/17/2024	If the \( M \times N \) matrix \( A \) has SVD \( A = U \Sigma V' \), then \( \mathcal{R}(A) = \mathcal{R}(U) \).A: True B: False
Published	10/17/2024	Example. Consider the symmetric outer product: \( A = zz' \), for \( z \neq 0 \). Because\[Az = (zz')z = z(z'z) = (z'z)z,\]where \( z'z \) is a scalar…
Published	10/17/2024	Example. Consider the symmetric outer product: \( A = zz' \), for \( z \neq 0 \). Because\[Az = (zz')z = z(z'z) = (z'z)z,\]where \( z'z \) is a scalar…
Published	10/17/2024	For Context:Example. Consider the symmetric outer product: \( A = zz' \), for \( z \neq 0 \). Because\[Az = (zz')z = z(z'z) = (z'z)z,\]where \( z'z \)…
Published	10/17/2024	For Context:Example. Consider the symmetric outer product: \( A = zz' \), for \( z \neq 0 \). Because\[Az = (zz')z = z(z'z) = (z'z)z,\]where \( z'z \)…
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