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Notes in
D. Uni::Linalg
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Status
Last Update
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Published
10/26/2024
Development of Linear Algebra
Published
10/26/2024
The abuse of notation
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10/26/2024
Special types of linear combinations
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10/26/2024
Cauchy-Schwarz-inequality
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10/26/2024
Triangle inequality
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10/26/2024
Scalar Rules
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10/26/2024
The angle between two vectors
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10/26/2024
Perpendicular vectors
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10/26/2024
Triangle inequality
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10/26/2024
Linear independence
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10/26/2024
Linear dependency in \(\mathbb{R}^2\)
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10/26/2024
Linear dependence (Alternative definiton)
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10/26/2024
Linear independence (Alternative definitions)
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10/26/2024
Uniqueness of linear combination
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10/26/2024
Span of vectors
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10/26/2024
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…
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10/26/2024
What's a matrix?
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10/26/2024
Two types of matrix notation
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10/26/2024
Matrix shapes
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10/26/2024
All types of square matrices
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10/26/2024
Column space
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10/26/2024
Rank
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10/26/2024
Let A be an \(m \times n \)matrix with \(r \) independent columns, and let \(C\) be the \(m \times r\) submatrix co…
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10/26/2024
Row space
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10/26/2024
Transpose
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10/26/2024
Matrix multiplication
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10/26/2024
Types of Matrix multiplication
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10/26/2024
Outer product rules
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10/26/2024
Matrix rules
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10/26/2024
CR decomposition
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10/26/2024
Matrix as a transformation
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10/26/2024
Definition of a linear transformation
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10/26/2024
The matrix of a linear transformation
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10/26/2024
Linear transformations and matrix multiplication
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10/26/2024
Systems of linear equations in the form \(Ax=b\)
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10/26/2024
Linear independency of columns in A when Ax=0 has a solution
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10/26/2024
Storing vectors and matrices in Arrays
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10/26/2024
How does gauss elimination work?
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10/26/2024
Prove solutions stay the same when applying Elimination or Permutation Matrices
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10/26/2024
Does applying a row operation change if A' has linearly independent columns?
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10/26/2024
What's the connection between Gauss elimination and the lineary independence of columns?
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10/26/2024
Gaussian elimination runtime
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10/26/2024
Definition of invertible matrices
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10/26/2024
Matrix Inverse formula for 2x2
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10/26/2024
Are inverses unique?
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10/26/2024
If \((AB)^{-1}\) is invertible then?
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10/26/2024
Invertiblity rules of transpose
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10/26/2024
The inverse theorem
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10/26/2024
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
10/26/2024
Proof \((ii) \implies (iii)\) of the inverse theorem:
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10/26/2024
Proof \((iii) \implies (ii)\) of the inverse theorem
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10/26/2024
Proof \((ii) \implies (i)\)
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10/26/2024
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
10/26/2024
Prove with two given \(m \times m\) matrices \(A\) and \(B\) with \(AB = I\) we have \(BA=I\)
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10/26/2024
Prove that a square lower triangular matrix is invertible if and only if all its diagonal entries are non-zero
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10/26/2024
Prove that the inverse of any lower triangular matrix, if it exists, is lower triangular itself
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10/26/2024
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
10/26/2024
Concept of LU-Decomposition
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10/26/2024
What can you do with LU decomposition?
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10/26/2024
The concept of LUP decomposition
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10/26/2024
Gauss-Jordan-elimination attributes
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10/26/2024
Finding the solution in Gauss-Jordan
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10/26/2024
Performing Gauss Jordan elimination
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10/26/2024
Gauss jordan and independent columns
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10/26/2024
CR-decomposition with Gauss Jordan
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10/26/2024
What's a vector space?
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10/26/2024
Examples of vector spaces
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10/26/2024
Prove V only contains one zero vector
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10/26/2024
Subpace definiton
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10/26/2024
Is the Columnspace C(A) a subspace of \(\mathbb{R}^m\)
Published
10/26/2024
Examples of more subspaces
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Last Update
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