kevinlui's site

Jan 24

• Introduce nullspace and columns space

• Draw picture of domain, range, kernel, codomain

Theorem related to linear independence

Let $S=\{a_1,\ldots,a_n\}\subseteq \mathbb{R}^m$ be a set of vectors. Let $A$ be the $m\times n$ matrix formed by writing the elements of $S$ as columns. Let $T:\mathbb{R}^n\to\mathbb{R}^m$ be the linear transformation defined by $T(x)=Ax$.

• $S$ is linearly independent

• $Ax=0$ has only the trivial solution

• For any $b$, $Ax=b$ has either no solution or exactly one solution.

• $null(A)=\{0\}$

• $T$ is one-to-one

• $ker(T)=\{0\}$

Theorem related to spanning

Let $S=\{a_1,\ldots,a_n\}\subseteq \mathbb{R}^m$ be a set of vectors. Let $A$ be the $m\times n$ matrix formed by writing the elements of $S$ as columns. Let $T:\mathbb{R}^n\to\mathbb{R}^m$ be the linear transformation defined by $T(x)=Ax$.

• $S$ is spanning

• $Ax=b$ has a solution for any $b$

• $col(A)=\mathbb{R}^m$

• $T$ is onto

• $range(T)=\mathbb{R}^m$

Theorem related to the square case

Let $S=\{a_1,\ldots,a_n\}\subseteq \mathbb{R}^n$ be a set of vectors. Let $A$ be the $n\times n$ matrix formed by writing the elements of $S$ as columns. Let $T:\mathbb{R}^n\to\mathbb{R}^n$ be the linear transformation defined by $T(x)=Ax$.

• $S$ is a basis

• $S$ is linearly independent

• $S$ is spanning

• $Ax=b$ always has a unique solution

• $col(A)=\mathbb{R}^n$

• $null(A)=\mathbb{R}^n$

• $T$ is invertible

• $A$ is invertible

3.2 Matrix Algebra

Matrix multiplication is weird

• $AB \neq BA$

• Explain what $AB=0$ means in terms of columnspace and nullspace

Tranpose of a matrix

• Teach how to tranpose

• $(A+B)^t = A^t + B^t$

• $(sA)^t$

• $(AC)^t = C^tA^t$

Diagonal matrices and upper triangular matrices is a thing

• Give definition

• The product of digaonl is diagonal. Discuss the effects of multiplying a matrice by a diagonal matrix

• The product of upper triangulars is upper triangular

Powers of matrices is a thing

• Powers of diagonal is easy

• Wouldn't it be great if $A=UDU^{-1}$

3.3 Inverses

• Explain what an inverse is.

• Derive inverse formula.