Notes in D. Uni::Linalg

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Status	Last Update	Fields
Published	11/13/2024	Development of Linear Algebra
Published	11/13/2024	The abuse of notation
Published	11/13/2024	Special types of linear combinations
Published	11/13/2024	Cauchy-Schwarz-inequality
Published	11/13/2024	Triangle inequality
Published	11/13/2024	Scalar Rules
Published	11/13/2024	The angle between two vectors
Published	11/13/2024	Perpendicular vectors
Published	11/13/2024	Triangle inequality
Published	11/13/2024	Linear independence
Published	11/13/2024	Linear dependency in \(\mathbb{R}^2\)
Published	11/13/2024	Linear dependence (Alternative definiton)
Published	11/13/2024	Linear independence (Alternative definitions)
Published	11/13/2024	Uniqueness of linear combination
Published	11/13/2024	Span of vectors
Published	11/13/2024	Let \(v_1, v_2, ...,v_n \in \mathbb{R}^m\) and let \(v \in \mathbb{R}^m\) be a linear combination of \(v_1, v_2, ...,v_n\)&nb…
Published	11/13/2024	What's a matrix?
Published	11/13/2024	Two types of matrix notation
Published	11/13/2024	Matrix shapes
Published	11/13/2024	All types of square matrices
Published	11/13/2024	Column space
Published	11/13/2024	Rank
Published	11/13/2024	Let A be an \(m \times n \)matrix with \(r \) independent columns, and let \(C\) be the \(m \times r\) submatrix co…
Published	11/13/2024	Row space
Published	11/13/2024	Transpose
Published	11/13/2024	Matrix multiplication
Published	11/13/2024	Types of Matrix multiplication
Published	11/13/2024	Outer product rules
Published	11/13/2024	Matrix rules
Published	11/13/2024	CR decomposition
Published	11/13/2024	Matrix as a transformation
Published	11/13/2024	Definition of a linear transformation
Published	11/13/2024	The matrix of a linear transformation
Published	11/13/2024	Linear transformations and matrix multiplication
Published	11/13/2024	Systems of linear equations in the form \(Ax=b\)
Published	11/13/2024	Linear independency of columns in A when Ax=0 has a solution
Published	11/13/2024	Storing vectors and matrices in Arrays
Published	11/13/2024	How does gauss elimination work?
Published	11/13/2024	Prove solutions stay the same when applying Elimination or Permutation Matrices
Published	11/13/2024	Does applying a row operation change if A' has linearly independent columns?
Published	11/13/2024	What's the connection between Gauss elimination and the lineary independence of columns?
Published	11/13/2024	Gaussian elimination runtime
Published	11/13/2024	Definition of invertible matrices
Published	11/13/2024	Matrix Inverse formula for 2x2
Published	11/13/2024	Are inverses unique?
Published	11/13/2024	If \((AB)^{-1}\) is invertible then?
Published	11/13/2024	Invertiblity rules of transpose
Published	11/13/2024	The inverse theorem
Published	11/13/2024	Proof\((i) \implies (ii)\) (i.e. that if A is invertible, then there's a solution \(Ax=b \) has a unique solution \(x \) for …
Published	11/13/2024	Proof \((ii) \implies (iii)\) of the inverse theorem:
Published	11/13/2024	Proof \((iii) \implies (ii)\) of the inverse theorem
Published	11/13/2024	Proof \((ii) \implies (i)\)
Published	11/13/2024	Let A and B be \(m \times m\) lower triangular matrices. Prove that AB is a lower triangular matrix. From this proof, prove that AB is …
Published	11/13/2024	Prove with two given \(m \times m\) matrices \(A\) and \(B\) with \(AB = I\) we have \(BA=I\)
Published	11/13/2024	Prove that a square lower triangular matrix is invertible if and only if all its diagonal entries are non-zero
Published	11/13/2024	Prove that the inverse of any lower triangular matrix, if it exists, is lower triangular itself
Published	11/13/2024	Assume we know that these two statements are true:(i) A square lower triangular matrix is invertible \(\iff\)all its diagonal entries are non-zer…
Published	11/13/2024	Concept of LU-Decomposition
Published	11/13/2024	What can you do with LU decomposition?
Published	11/13/2024	The concept of LUP decomposition
Published	11/13/2024	Gauss-Jordan-elimination attributes
Published	11/13/2024	Finding the solution in Gauss-Jordan
Published	11/13/2024	Performing Gauss Jordan elimination
Published	11/13/2024	Gauss jordan and independent columns
Published	11/13/2024	CR-decomposition with Gauss Jordan
Published	11/13/2024	What's a vector space?
Published	11/13/2024	Examples of vector spaces
Published	11/13/2024	Prove V only contains one zero vector
Published	11/13/2024	Subpace definiton
Published	11/13/2024	Is the Columnspace C(A) a subspace of \(\mathbb{R}^m\)
Published	11/13/2024	Examples of more subspaces
Published	11/13/2024	What's a basis?
Published	11/13/2024	Let V be a vector space, \(G \subseteq V \). A linear combination of G is of the form...
Published	11/13/2024	Proof, that every linear combination of \(G \subseteq V\) is again in V
Published	11/13/2024	If you take the infinite linear combination of \(G \subseteq V\), is it still in \(V\)?
Published	11/13/2024	Let \(V \) be a vector space \(G \subseteq\)V is a subset of vectors. Definiton of Span(G) and when is \(G\) linearly indepen…
Published	11/13/2024	Formal definiton basis
Published	11/13/2024	Can there only be one base?
Published	11/13/2024	Steinitz exchange lemma:
Published	11/13/2024	Proof of Steinitz exchange lemma
Published	11/13/2024	Proof: Let \(V \) be a vector space, \(B,B'\subseteq V\)two finite bases of V. Then \(\|B\|=\|B'\|\)
Published	11/13/2024	Does every vector space have a basis?finite case vs infinite case
Published	11/13/2024	Dimension of a vector space
Published	11/13/2024	Simplified basis criterion
Published	11/13/2024	Fundamental subspaces of a matrix A and how to compute:
Published	11/13/2024	Column space of a matrix A, proof Gauss Jordan
Published	11/13/2024	Is \(R(A)\) a subspace of \(\mathbb{R}^m\)
Published	11/13/2024	Proof \(R(A)=R(MA) \) if M is invertible
Published	11/13/2024	Proof: \(dim(R(A))=r=rank(A)\)
Published	11/13/2024	Does row rank equal column space?
Published	11/13/2024	Definiton nullspace
Published	11/13/2024	How to get the basis of N(A)
Published	11/13/2024	Dimension of Nullspace compared to other spaces
Published	11/13/2024	Solution space definition and basis
Published	11/13/2024	Relevance of orthogonality of vectors and suspaces
Published	11/13/2024	Definition orthognality
Published	11/13/2024	Orthogonality of subspaces proof
Published	11/13/2024	The basis of two orthogonal subspaces is linear independent, Proof
Published	11/13/2024	Definiton of orthogonal subspaces
Published	11/13/2024	The nullspace is the orthogonal complent of the row space, proof
Published	11/13/2024	Orthongal subspaces, alternate defintions
Published	11/13/2024	Decomposition of \(\mathbb{R}^n\)
Published	11/13/2024	Set of all solutions to a system of linear equations
Published	11/13/2024	Link between nullspaces of A and \(A^\intercal A\)
Published	11/13/2024	Projection of a vector onto a subspace
Published	11/13/2024	Projection onto a line and proof of the formula
Published	11/13/2024	Generlation of projection onto non one dimensional space and proof
Status	Last Update	Fields