Introduction to Matrices

Matrix Representations of Linear Maps

A matrix is a rectangular array of scalars arranged in rows and columns. Matrices can be used to represent linear maps between finite-dimensional vector spaces.

A matrix with $𝑚$ rows and $𝑛$ columns with elements from a field $𝐹$ is called an $𝑚 \times 𝑛$ matrix over $𝐹$ .

Notation: Sets of Matrices

There are various ways to denote the set of all $𝑚 \times 𝑛$ matrices over $𝐹$ , most commonly:

ℳ_{𝑚 \times 𝑛} (𝐹), 𝐹^{𝑚 \times 𝑛} .

If $𝐴$ is a matrix, then $𝑎_{𝑖 𝑗}$ denotes the element in the $𝑖$ ^th row and $𝑗$ ^th column of $𝐴$ :

𝐴 = (\begin{matrix} 𝑎_{11} & 𝑎_{12} & \dots & 𝑎_{1 𝑛} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑚 1} & 𝑎_{𝑚 2} & \dots & 𝑎_{𝑚 𝑛} \end{matrix}) .

Notation: Column and Row Vectors

A column vector represents a vector in $𝐹^{𝑚}$ as an $𝑚 \times 1$ matrix, and a row vector represents a vector in $𝐹^{𝑛}$ as a $1 \times 𝑛$ matrix.

Example.

The vector

𝐯 = (1, 2, 3) \in ℝ^{3}

can be represented as the column vector

(\begin{matrix} 1 \\ 2 \\ 3 \end{matrix})

or the row vector

(\begin{matrix} 1 & 2 & 3 \end{matrix})

Despite the abstract notion of linear maps, by choosing bases for the domain and codomain vector spaces, we can represent linear maps concretely as matrices in familiar fields like $ℝ$ and $ℂ$ .

Definition 33: Standard Matrix of a Linear Map

Let $𝑉$ and $𝑊$ be finite-dimensional vector spaces over the same field $𝐹$ , and let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Let $ℬ = {𝐛_{1}, 𝐛_{2}, \dots, 𝐛_{𝑛}}$ be an ordered basis for $𝑉$ , and let $𝒞 = {𝐜_{1}, 𝐜_{2}, \dots, 𝐜_{𝑚}}$ be an ordered basis for $𝑊$ .

The standard matrix of $𝑇$ with respect to the bases $ℬ$ and $𝒞$ , denoted ${[𝑇]}_{𝒞 ℬ}$ , is the $𝑚 \times 𝑛$ matrix whose $𝑖$ ^th column is the coordinate vector of $𝑇 (𝐛_{𝑖})$ with respect to the basis $𝒞$ , where $𝐛_{𝑖}$ is the $𝑖$ ^th basis vector in $ℬ$ :

{[𝑇]}_{𝒞 ℬ} = (\begin{matrix} | & | & | \\ {[𝑇 (𝐛_{1})]}_{𝒞} & {[𝑇 (𝐛_{2})]}_{𝒞} & \dots & {[𝑇 (𝐛_{𝑛})]}_{𝒞} \\ | & | & | \end{matrix}) .

Corollary 15: Dimension of a Standard Matrix

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map between finite-dimensional vector spaces. If $\dim (𝑉) = 𝑛$ and $\dim (𝑊) = 𝑚$ , then any standard matrix for $𝑇$ is an $𝑚 \times 𝑛$ matrix.

Example.

Let $ℙ_{3} (ℝ)$ be the real vector space of all polynomials of degree at most 3 over the indeterminate $𝑥$ with real coefficients. Also, consider the standard polynomial basis $ℬ = {1, 𝑥, 𝑥^{2}, 𝑥^{3}}$ for $ℙ_{3} (ℝ)$ .

Recall that differentiation is a linear map, so define $𝐷 : ℙ_{3} (ℝ) \to ℙ_{3} (ℝ)$ by $𝐷 (𝑓) = d 𝑓 / d 𝑥$ . The standard matrix of $𝐷$ with respect to the standard polynomial basis $ℬ$ is the $4 \times 4$ matrix:

\begin{matrix} {[𝐷]}_{ℬ ℬ} & = (\begin{matrix} | & | & | & | \\ {[𝐷 (1)]}_{ℬ} & {[𝐷 (𝑥)]}_{ℬ} & {[𝐷 (𝑥^{2})]}_{ℬ} & {[𝐷 (𝑥^{3})]}_{ℬ} \\ | & | & | & | \end{matrix}) \\ = (\begin{matrix} | & | & | & | \\ {[0]}_{ℬ} & {[1]}_{ℬ} & {[2 𝑥]}_{ℬ} & {[3 𝑥^{2}]}_{ℬ} \\ | & | & | & | \end{matrix}) = (\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{matrix}) . \end{matrix}

Example.

Let $𝑇 : ℝ^{2} \to ℝ^{3}$ be a linear map defined by $𝑇 (𝑥, 𝑦) = (𝑥, 3 𝑥 + 𝑦, 𝑥 - 2 𝑦)$ . The standard matrix of $𝑇$ with respect to the standard bases for $ℝ^{2}$ and $ℝ^{3}$ ( $ℰ_{2}$ and $ℰ_{3}$ respectively) is the $3 \times 2$ matrix:

\begin{matrix} {[𝑇]}_{ℰ_{3} ℰ_{2}} & = (\begin{matrix} | & | \\ {[𝑇 (𝐞_{1})]}_{ℰ_{3}} & {[𝑇 (𝐞_{2})]}_{ℰ_{3}} \\ | & | \end{matrix}) \\ = (\begin{matrix} | & | \\ 𝑇 (𝐞_{1}) & 𝑇 (𝐞_{2}) \\ | & | \end{matrix}) because {[𝐯]}_{ℰ} = 𝐯 \\ = (\begin{matrix} | & | \\ 𝑇 (1, 0) & 𝑇 (0, 1) \\ | & | \end{matrix}) = (\begin{matrix} 1 & 0 \\ 3 & 1 \\ 1 & - 2 \end{matrix}) . \end{matrix}

In general, if $𝑇 : 𝑉 \to 𝐹^{𝑚}$ is a linear map, we can use the standard basis $ℰ_{𝑚}$ for $𝐹^{𝑚}$ to construct a standard matrix for $𝑇$ without needing to compute coordinate vectors. Since ${[𝐯]}_{ℰ_{𝑚}} = 𝐯$ for all $𝐯 \in 𝐹^{𝑚}$ :

{[𝑇]}_{ℰ_{𝑚} ℬ} = (\begin{matrix} | & | & | \\ 𝑇 (𝐛_{1}) & 𝑇 (𝐛_{2}) & \dots & 𝑇 (𝐛_{𝑛}) \\ | & | & | \end{matrix}) where ℬ = {𝐛_{1}, 𝐛_{2}, \dots, 𝐛_{𝑛}} is a basis for 𝑉 .

Furthermore, if $𝑉 = 𝐹^{𝑛}$ , we can use the standard basis $ℰ_{𝑛}$ for $𝐹^{𝑛}$ to construct a standard matrix for $𝑇$ based on only the standard basis vectors $𝐞_{1}, 𝐞_{2}, \dots, 𝐞_{𝑛}$ :

{[𝑇]}_{ℰ_{𝑚} ℰ_{𝑛}} = (\begin{matrix} | & | & | \\ 𝑇 (𝐞_{1}) & 𝑇 (𝐞_{2}) & \dots & 𝑇 (𝐞_{𝑛}) \\ | & | & | \end{matrix}) where ℰ_{𝑛} = {𝐞_{1}, 𝐞_{2}, \dots, 𝐞_{𝑛}} .

This specific standard matrix of $𝑇$ is often simply called the standard matrix or coordinate matrix of $𝑇$ , denoted $[𝑇]$ .

Corollary 16: Standard Matrix of a Linear Map from $𝐹^{𝑛}$ to $𝐹^{𝑚}$

Let $𝑇 : 𝐹^{𝑛} \to 𝐹^{𝑚}$ be a linear map. Then the standard matrix of $𝑇$ , denoted $[𝑇]$ , with respect to the standard bases for $𝐹^{𝑛}$ and $𝐹^{𝑚}$ ( $ℰ_{𝑛}$ and $ℰ_{𝑚}$ respectively) is the $𝑚 \times 𝑛$ matrix whose $𝑖$ ^th column is the image of $𝐞_{𝑖} \in 𝐹^{𝑛}$ under $𝑇$ . That is,

[𝑇] = {[𝑇]}_{ℰ_{𝑚} ℰ_{𝑛}} = (\begin{matrix} | & | & | \\ 𝑇 (𝐞_{1}) & 𝑇 (𝐞_{2}) & \dots & 𝑇 (𝐞_{𝑛}) \\ | & | & | \end{matrix}) .

Now suppose $𝐯 \in 𝑉$ and $𝑇 : 𝑉 \to 𝑊$ is a linear map. Also let $ℬ$ and $𝒞$ be bases for $𝑉$ and $𝑊$ respectively.

Definition 34: Matrix-Vector Product

Suppose $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ has columns $𝐚_{1}, 𝐚_{2}, \dots, 𝐚_{𝑛} \in 𝐹^{𝑚}$ , and let $𝐱 = (𝑥_{1}, 𝑥_{2}, \dots, 𝑥_{𝑛}) \in 𝐹^{𝑛}$ be a column vector. That is:

𝐴 = (\begin{matrix} | & | & | \\ 𝐚_{1} & 𝐚_{2} & \dots & 𝐚_{𝑛} \\ | & | & | \end{matrix}), 𝐱 = (\begin{matrix} 𝑥_{1} \\ ⋮ \\ 𝑥_{𝑛} \end{matrix}) .

The matrix-vector product of $𝐴$ and $𝐱$ , denoted $𝐴 𝐱$ , is the vector in $𝐹^{𝑚}$ defined by:

𝐴 𝐱 = 𝑥_{1} 𝐚_{1} + 𝑥_{2} 𝐚_{2} + \dots + 𝑥_{𝑛} 𝐚_{𝑛} .

In other words, $𝐴 𝐱$ is the linear combination of the columns of $𝐴$ whose weights are given by the entries of $𝐱$ .

Example.

Let $𝐴 = (\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix})$ and $𝐱 = (\begin{matrix} 5 \\ 6 \end{matrix})$ . Then the matrix-vector product of $𝐴$ and $𝐱$ is:

𝐴 𝐱 = 5 (\begin{matrix} 1 \\ 3 \end{matrix}) + 6 (\begin{matrix} 2 \\ 4 \end{matrix}) = (\begin{matrix} 17 \\ 39 \end{matrix}) .

Theorem 35: Fundamental Property of the Matrix-Vector Product

The matrix-vector product of the standard matrix ${[𝑇]}_{𝒞 ℬ}$ of $𝑇$ with respect to the bases $ℬ$ and $𝒞$ and the coordinate vector ${[𝐯]}_{ℬ}$ of $𝐯$ with respect to the basis $ℬ$ is:

{[𝑇]}_{𝒞 ℬ} {[𝐯]}_{ℬ} = {[𝑇 (𝐯)]}_{𝒞} .

Proof: Theorem 35.

Suppose $𝑇 : 𝑉 \to 𝑊$ is a linear map. Let $ℬ = {𝐛_{1}, 𝐛_{2}, \dots, 𝐛_{𝑛}}$ be an ordered basis for $𝑉$ , and let $𝒞 = {𝐜_{1}, 𝐜_{2}, \dots, 𝐜_{𝑚}}$ be an ordered basis for $𝑊$ . Let $𝐯 \in 𝑉$ be a vector with $ℬ$ -coordinates ${[𝐯]}_{ℬ} = (𝑥_{1}, 𝑥_{2}, \dots, 𝑥_{𝑛})$ , such that $𝐯 = 𝑥_{1} 𝐛_{1} + 𝑥_{2} 𝐛_{2} + \dots + 𝑥_{𝑛} 𝐛_{𝑛}$ . Then:

\begin{matrix} {[𝑇]}_{𝒞 ℬ} {[𝐯]}_{ℬ} & = (\begin{matrix} | & | \\ {[𝑇 (𝐛_{1})]}_{𝒞} & \dots & {[𝑇 (𝐛_{𝑛})]}_{𝒞} \\ | & | \end{matrix}) (\begin{matrix} 𝑥_{1} \\ ⋮ \\ 𝑥_{𝑛} \end{matrix}) \\ = 𝑥_{1} {[𝑇 (𝐛_{1})]}_{𝒞} + 𝑥_{2} {[𝑇 (𝐛_{2})]}_{𝒞} + \dots + 𝑥_{𝑛} {[𝑇 (𝐛_{𝑛})]}_{𝒞} & by Definition 34 \\ = {[𝑥_{1} 𝑇 (𝐛_{1}) + 𝑥_{2} 𝑇 (𝐛_{2}) + \dots + 𝑥_{𝑛} 𝑇 (𝐛_{𝑛})]}_{𝒞} & by Theorem 29 \\ = {[𝑇 (𝑥_{1} 𝐛_{1} + 𝑥_{2} 𝐛_{2} + \dots + 𝑥_{𝑛} 𝐛_{𝑛})]}_{𝒞} & because 𝑇 is linear \\ = {[𝑇 (𝐯)]}_{𝒞} & by substitution of 𝐯, \end{matrix}

as desired. ∎

Definition 35: Identity Matrix

The $𝒏 \times 𝒏$ identity matrix, written $𝐼_{𝑛}$ , is the $𝑛 \times 𝑛$ standard matrix of the identity map ${id}_{𝐹^{𝑛}}$ with respect to the standard basis $ℰ_{𝑛} = {𝐞_{1}, 𝐞_{2}, \dots, 𝐞_{𝑛}}$ :

𝐼_{𝑛} = {[{id}_{𝐹^{𝑛}}]}_{ℰ_{𝑛} ℰ_{𝑛}} = (\begin{matrix} | & | & | \\ 𝐞_{1} & 𝐞_{2} & \dots & 𝐞_{𝑛} \\ | & | & | \end{matrix}) = (\begin{matrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{matrix}) .

Corollary 17

Let $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ . Then $𝐼_{𝑚} 𝐴 = 𝐴 𝐼_{𝑛} = 𝐴$ .

Corollary 18

Let $𝐯 \in 𝐹^{𝑛}$ . Then $𝐼_{𝑛} 𝐯 = 𝐯$ .

Definition 36: Standard Transformation of a Matrix

The standard transformation of a matrix $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ is the linear map $𝑇_{𝐴} : 𝐹^{𝑛} \to 𝐹^{𝑚}$ defined by:

𝑇_{𝐴} (𝐱) = 𝐴 𝐱 for all 𝐱 \in 𝐹^{𝑛} .

Proposition 7

Let $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ be a matrix and let $𝑇_{𝐴} : 𝐹^{𝑛} \to 𝐹^{𝑚}$ be the standard transformation of $𝐴$ . Then the standard matrix of $𝑇_{𝐴}$ with respect to the standard bases for $𝐹^{𝑛}$ and $𝐹^{𝑚}$ is $𝐴$ , i.e. $[𝑇_{𝐴}] = 𝐴$ .

Operations on Matrices

Definition 37: Matrix Multiplication

Let $𝐴 \in 𝐹^{𝑚 \times 𝑝}$ and $𝐵 \in 𝐹^{𝑝 \times 𝑛}$ be matrices. The matrix product of $𝐴$ and $𝐵$ , denoted $𝐴 𝐵$ , is the $𝑚 \times 𝑛$ matrix whose $𝑖$ ^th column is the matrix-vector product of $𝐴$ and the $𝑖$ ^th column of $𝐵$ :

𝐴 𝐵 = 𝐴 (\begin{matrix} 𝐛_{1} & 𝐛_{2} & \dots & 𝐛_{𝑛} \end{matrix}) = (\begin{matrix} 𝐴 𝐛_{1} & 𝐴 𝐛_{2} & \dots & 𝐴 𝐛_{𝑛} \end{matrix}) .

Theorem 36: Fundamental Property of Matrix Multiplication

Let $𝑈, 𝑉, 𝑊$ be finite-dimensional vector spaces over a field $𝐹$ , and let $𝑆 : 𝑉 \to 𝑊$ and $𝑇 : 𝑈 \to 𝑉$ be linear maps. Let $ℬ$ , $𝒞$ , and $𝒟$ be ordered bases for $𝑈$ , $𝑉$ , and $𝑊$ respectively. Then:

{[𝑆]}_{𝒟 𝒞} {[𝑇]}_{𝒞 ℬ} = {[𝑆 \circ 𝑇]}_{𝒟 ℬ} .

Proof: Theorem 36.

Let $𝑆 : 𝑉 \to 𝑊$ and $𝑇 : 𝑈 \to 𝑉$ be linear maps between finite-dimensional vector spaces, and let $ℬ = {𝐛_{1}, \dots, 𝐛_{𝑛}}$ , $𝒞$ , and $𝒟$ be ordered bases for $𝑈$ , $𝑉$ , and $𝑊$ respectively. Then:

\begin{matrix} {[𝑆]}_{𝒟 𝒞} {[𝑇]}_{𝒞 ℬ} & = {[𝑆]}_{𝒟 𝒞} (\begin{matrix} | & | \\ {[𝑇 (𝐛_{1})]}_{𝒞} & \dots & {[𝑇 (𝐛_{𝑛})]}_{𝒞} \\ | & | \end{matrix}) & by Definition 33 \\ = (\begin{matrix} | & | \\ {[𝑆]}_{𝒟 𝒞} {[𝑇 (𝐛_{1})]}_{𝒞} & \dots & {[𝑆]}_{𝒟 𝒞} {[𝑇 (𝐛_{𝑛})]}_{𝒞} \\ | & | \end{matrix}) & by Definition 37 \\ = (\begin{matrix} | & | \\ {[𝑆 (𝑇 (𝐛_{1}))]}_{𝒟} & \dots & {[𝑆 (𝑇 (𝐛_{𝑛}))]}_{𝒟} \\ | & | \end{matrix}) & by Theorem 35 \\ = (\begin{matrix} | & | \\ {[(𝑆 \circ 𝑇) (𝐛_{1})]}_{𝒟} & \dots & {[(𝑆 \circ 𝑇) (𝐛_{𝑛})]}_{𝒟} \\ | & | \end{matrix}) & by definition of function composition \\ = {[𝑆 \circ 𝑇]}_{𝒟 ℬ} & by Definition 33, \end{matrix}

as desired. ∎

Theorem 37: Alternative Formula for Matrix Multiplication

Let $𝐴 \in 𝐹^{𝑚 \times 𝑝}$ and $𝐵 \in 𝐹^{𝑝 \times 𝑛}$ be matrices. Denote $𝐚_{𝑖} \in 𝐹^{𝑝}$ as the $𝑖$ ^th row of $𝐴$ and $𝐛_{𝑗} \in 𝐹^{𝑝}$ as the $𝑗$ ^th column of $𝐵$ .

Define a function

¹ $𝑓 : 𝐹^{𝑝} \times 𝐹^{𝑝} \to 𝐹$ by $𝑓 ((𝑥_{1}, \dots, 𝑥_{𝑝}), (𝑦_{1}, \dots, 𝑦_{𝑝})) = \sum_{𝑘 = 1}^{𝑝} 𝑥_{𝑘} 𝑦_{𝑘}$ . Then the $(𝑖, 𝑗)$ ^th entry of the matrix product $𝐴 𝐵$ is $𝑓 (𝐚_{𝑖}, 𝐛_{𝑗})$ :

(\begin{matrix} ― & 𝐚_{1} & ― \\ ― & 𝐚_{2} & ― \\ ⋮ \\ ― & 𝐚_{𝑚} & ― \end{matrix}) (\begin{matrix} | & | & | \\ 𝐛_{1} & 𝐛_{2} & \dots & 𝐛_{𝑛} \\ | & | & | \end{matrix}) = (\begin{matrix} 𝑓 (𝐚_{1}, 𝐛_{1}) & 𝑓 (𝐚_{1}, 𝐛_{2}) & \dots & 𝑓 (𝐚_{1}, 𝐛_{𝑛}) \\ 𝑓 (𝐚_{2}, 𝐛_{1}) & 𝑓 (𝐚_{2}, 𝐛_{2}) & \dots & 𝑓 (𝐚_{2}, 𝐛_{𝑛}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑓 (𝐚_{𝑚}, 𝐛_{1}) & 𝑓 (𝐚_{𝑚}, 𝐛_{2}) & \dots & 𝑓 (𝐚_{𝑚}, 𝐛_{𝑛}) \end{matrix}) .

Definition 38: Transpose of a Matrix

Let $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ be a matrix. The transpose of $𝐴$ , denoted $𝐴^{𝖳}$ , is the $𝑛 \times 𝑚$ matrix obtained by interchanging the rows and columns of $𝐴$ :

𝐴^{𝖳} = (\begin{matrix} 𝑎_{11} & 𝑎_{21} & \dots & 𝑎_{𝑚 1} \\ 𝑎_{12} & 𝑎_{22} & \dots & 𝑎_{𝑚 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{1 𝑛} & 𝑎_{2 𝑛} & \dots & 𝑎_{𝑚 𝑛} \end{matrix}) .

Example.

The transpose of a $2 \times 3$ matrix is a $3 \times 2$ matrix:

{(\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix})}^{𝖳} = (\begin{matrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{matrix}) .

Definition 39: Matrix Addition and Scalar Multiplication

Let $𝐴, 𝐵 \in 𝐹^{𝑚 \times 𝑛}$ be matrices. The matrix addition of $𝐴$ and $𝐵$ , denoted $𝐴 + 𝐵$ , is the $𝑚 \times 𝑛$ matrix obtained by adding corresponding entries of $𝐴$ and $𝐵$ :

𝐴 + 𝐵 = (\begin{matrix} 𝑎_{11} + 𝑏_{11} & 𝑎_{12} + 𝑏_{12} & \dots & 𝑎_{1 𝑛} + 𝑏_{1 𝑛} \\ 𝑎_{21} + 𝑏_{21} & 𝑎_{22} + 𝑏_{22} & \dots & 𝑎_{2 𝑛} + 𝑏_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑚 1} + 𝑏_{𝑚 1} & 𝑎_{𝑚 2} + 𝑏_{𝑚 2} & \dots & 𝑎_{𝑚 𝑛} + 𝑏_{𝑚 𝑛} \end{matrix}) .

Let $𝑐 \in 𝐹$ be a scalar. The scalar multiplication of $𝐴$ by $𝑐$ , denoted $𝑐 𝐴$ , is the matrix obtained by multiplying each entry of $𝐴$ by the scalar $𝑐$ :

𝑐 𝐴 = (\begin{matrix} 𝑐 \cdot 𝑎_{11} & 𝑐 \cdot 𝑎_{12} & \dots & 𝑐 \cdot 𝑎_{1 𝑛} \\ 𝑐 \cdot 𝑎_{21} & 𝑐 \cdot 𝑎_{22} & \dots & 𝑐 \cdot 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑐 \cdot 𝑎_{𝑚 1} & 𝑐 \cdot 𝑎_{𝑚 2} & \dots & 𝑐 \cdot 𝑎_{𝑚 𝑛} \end{matrix}) .

Theorem 38: $𝐹^{𝑚 \times 𝑛}$ is a Vector Space

$𝐹^{𝑚 \times 𝑛}$ , the set of all $𝑚 \times 𝑛$ matrices over the field $𝐹$ , is an $𝑚 𝑛$ -dimensional vector space over $𝐹$ under the matrix addition and scalar multiplication as defined in Definition 39.

Proof: Theorem 38.

Suppose $𝐹$ is an arbitrary field, and $𝑚, 𝑛 \in ℤ^{+}$ .

Lemma 1:

𝐹^{𝑚 \times 𝑛}

is a vector space over

𝐹

Lemma 2:

\dim (𝐹^{𝑚 \times 𝑛}) = 𝑚 𝑛

Thus, $𝐹^{𝑚 \times 𝑛}$ is an $𝑚 𝑛$ -dimensional vector space over $𝐹$ .

Subspaces Defined by Matrices

Definition 40: Column Space of a Matrix

Let $𝐴$ be an $𝑚 \times 𝑛$ matrix over a field $𝐹$ . The column space of $𝐴$ , denoted $col (𝐴)$ , is the subspace of $𝐹^{𝑚}$ spanned by the columns of $𝐴$ :

col (\begin{matrix} | & | & | \\ 𝐚_{1} & 𝐚_{2} & \dots & 𝐚_{𝑛} \\ | & | & | \end{matrix}) = span {𝐚_{1}, 𝐚_{2}, \dots, 𝐚_{𝑛}} .

Definition 41: Row Space of a Matrix

Let $𝐴$ be an $𝑚 \times 𝑛$ matrix over a field $𝐹$ . The row space of $𝐴$ , denoted $row (𝐴)$ , is the subspace of $𝐹^{𝑛}$ spanned by the rows of $𝐴$ :

row (\begin{matrix} ― & 𝐚_{1} & ― \\ ― & 𝐚_{2} & ― \\ ⋮ \\ ― & 𝐚_{𝑚} & ― \end{matrix}) = span {𝐚_{1}, 𝐚_{2}, \dots, 𝐚_{𝑚}} .

Important

Suppose $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ . Then $col (𝐴)$ is a subspace of $𝐹^{𝑚}$ and $row (𝐴)$ is a subspace of $𝐹^{𝑛}$ .

Corollary 19

$col (𝐴) = row (𝐴^{𝖳})$ , and $row (𝐴) = col (𝐴^{𝖳})$ .

Corollary 20

$col (𝐴) = im (𝑇_{𝐴})$ , where $𝑇_{𝐴} : 𝐹^{𝑛} \to 𝐹^{𝑚}$ is defined by $𝐱 \mapsto 𝐴 𝐱$ .

Definition 42: Null Space of a Matrix

Let $𝐴$ be an $𝑚 \times 𝑛$ matrix over a field $𝐹$ . The null space of $𝐴$ , denoted $nul (𝐴)$ , is the subspace of $𝐹^{𝑛}$ defined by:

nul (𝐴) = {𝐱 \in 𝐹^{𝑛} | 𝐴 𝐱 = 𝟎} .

Important

Suppose $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ . Then $nul (𝐴)$ is a subspace of $𝐹^{𝑛}$ .

Corollary 21

$nul (𝐴) = \ker (𝑇_{𝐴})$ , where $𝑇_{𝐴} : 𝐹^{𝑛} \to 𝐹^{𝑚}$ is defined by $𝐱 \mapsto 𝐴 𝐱$ .

Definition 43: Rank and Nullity of a Matrix

Let $𝐴$ be an $𝑚 \times 𝑛$ matrix over a field $𝐹$ . The rank of $𝐴$ , denoted $rank (𝐴)$ , is the dimension of the column space of $𝐴$ :

rank (𝐴) = \dim (col (𝐴)) .

The nullity of $𝐴$ , denoted $nullity (𝐴)$ , is the dimension of the null space of $𝐴$ :

nullity (𝐴) = \dim (nul (𝐴)) .

Corollary 22

$\dim (col (𝐴)) = \dim (row (𝐴)) = rank (𝐴)$ .

Theorem 39: Rank-Nullity Theorem for Matrices

Let $𝐴$ be an $𝑚 \times 𝑛$ matrix over a field $𝐹$ . Then:

rank (𝐴) + nullity (𝐴) = 𝑛 .

Exercises

Exercise 62.

Let $𝐴 \in 𝐹^{𝑚 \times 𝑘}$ and $𝐵 \in 𝐹^{𝑘 \times 𝑛}$ . Show that:

rank (𝐴 𝐵) \leq \min (rank (𝐴), rank (𝐵)) .

Exercise 63.

Suppose

𝐴, 𝐵 \in 𝐹^{𝑛 \times 𝑛}

with

𝐴^{2} = 𝐵^{2}

and

𝐴^{3} = 𝐵^{3}

. Prove or disprove that necessarily

𝐴 = 𝐵

Exercise 64.

Suppose

𝑉

and

𝑊

are finite-dimensional nontrivial vector spaces, and

𝑇 : 𝑉 \to 𝑊

is a linear map. Show that

rank (𝑇) = 1

if and only if there exists a basis

ℬ

for

𝑉

and a basis

𝒞

for

𝑊

such that all entries of

{[𝑇]}_{𝒞 ℬ}

equal

1

Exercise 65.

Let $ℙ_{𝑛} (ℝ)$ denote the vector space of all polynomials of degree at most $𝑛$ over the indeterminate $𝑥$ with real coefficients. Let $ℬ_{𝑛} = {1, 𝑥, 𝑥^{2}, \dots, 𝑥^{𝑛}}$ be the standard polynomial basis for $ℙ_{𝑛} (ℝ)$ .

Define the integration map $𝐽 : ℙ_{3} (ℝ) \to ℙ_{4} (ℝ)$ on polynomials of at most degree 3 by:

𝐽 (𝑓) = \int_{0}^{𝑥} 𝑓 (𝑡) d 𝑡 .

Show that $𝐽$ is a linear map and find ${[𝐽]}_{ℬ_{4} ℬ_{3}}$ , the $5 \times 4$ standard matrix of $𝐽$ with respect to the standard polynomial bases for $ℙ_{3} (ℝ)$ and $ℙ_{4} (ℝ)$ .

¹This is the dot product, which is redefined later