Invertibility and Isomorphisms

Inverses of Linear Maps and Matrices

Definition 44: Inverse of a Linear Map

Let $𝑉$ and $𝑊$ be vector spaces over a field $𝐹$ . A linear map $𝑇 : 𝑉 \to 𝑊$ is invertible iff there exists a linear map $𝑇^{- 1} : 𝑊 \to 𝑉$ , called the inverse of $𝑇$ , such that $𝑇^{- 1} \circ 𝑇 = {id}_{𝑉}$ and $𝑇 \circ 𝑇^{- 1} = {id}_{𝑊}$ , where ${id}_{𝑉}$ and ${id}_{𝑊}$ are the identity maps on $𝑉$ and $𝑊$ , respectively.

Theorem 40: Bijections are Invertible

A linear map $𝑇 : 𝑉 \to 𝑊$ is invertible if and only if $𝑇$ is bijective (injective and surjective).

Theorem 41: Dimension of Invertible Linear Maps

Let $𝑉$ and $𝑊$ be finite-dimensional vector spaces over a field $𝐹$ . Then if any linear map $𝑇 : 𝑉 \to 𝑊$ is invertible, $\dim (𝑉) = \dim (𝑊)$ .

Theorem 42: Equivalence of Injectivity, Surjectivity, and Invertibility

Suppose $𝑉$ and $𝑊$ are finite-dimensional vector spaces with $\dim (𝑉) = \dim (𝑊)$ . For all linear maps $𝑇 : 𝑉 \to 𝑊$ , $𝑇$ is invertible iff $𝑇$ is injective iff $𝑇$ is surjective.

Definition 45: Inverse of a Matrix

Let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ be a square matrix. $𝐴$ is invertible iff there exists a square matrix $𝐴^{- 1} \in 𝐹^{𝑛 \times 𝑛}$ , called the inverse of $𝐴$ , such that $𝐴^{- 1} 𝐴 = 𝐼_{𝑛}$ and $𝐴 𝐴^{- 1} = 𝐼_{𝑛}$ .

Theorem 43: Inverse of a $2 \times 2$ Matrix

Let $𝐴 = (\begin{matrix} 𝑎 & 𝑏 \\ 𝑐 & 𝑑 \end{matrix}) \in 𝐹^{2 \times 2}$ . Then $𝐴$ is invertible if and only if $𝑎 𝑑 - 𝑏 𝑐 \neq 0$ , in which case the inverse of $𝐴$ is given by:

𝐴^{- 1} = \frac{1}{𝑎 𝑑 - 𝑏 𝑐} (\begin{matrix} 𝑑 & - 𝑏 \\ - 𝑐 & 𝑎 \end{matrix}) .

Definition 46: Left and Right Inverses of Linear Maps

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map.

A linear map $𝑆 : 𝑊 \to 𝑉$ is a left inverse of $𝑇$ iff $𝑆 \circ 𝑇 = {id}_{𝑉}$ .
A linear map $𝑅 : 𝑊 \to 𝑉$ is a right inverse of $𝑇$ iff $𝑇 \circ 𝑅 = {id}_{𝑊}$ .

Definition 47: Left and Right Inverses of Matrices

Let $𝐴 \in 𝐹^{𝑚 \times 𝑛}$ be a matrix.

A matrix $𝐵 \in 𝐹^{𝑛 \times 𝑚}$ is a left inverse of $𝐴$ iff $𝐵 𝐴 = 𝐼_{𝑛}$ .
A matrix $𝐶 \in 𝐹^{𝑛 \times 𝑚}$ is a right inverse of $𝐴$ iff $𝐴 𝐶 = 𝐼_{𝑚}$ .

Corollary 23

A linear map or matrix is invertible if and only if it has both a left and right inverse.

Proposition 8: Uniqueness of Inverses

Let $𝑇 : 𝑉 \to 𝑊$ be an invertible linear map. Then the inverse of $𝑇$ is unique. Similarly, let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ be an invertible matrix. Then the inverse of $𝐴$ is unique.

Theorem 44: Matrix of Inverse Equals Inverse of Matrix

Let $𝑇 : 𝑉 \to 𝑊$ be an invertible linear map, and let $ℬ$ and $𝒞$ be bases for $𝑉$ and $𝑊$ , respectively. Then the standard matrix of the inverse of $𝑇$ is the inverse of the standard matrix of $𝑇$ :

{[𝑇^{- 1}]}_{ℬ 𝒞} = {({[𝑇]}_{𝒞 ℬ})}^{- 1} .

Isomorphisms

Definition 48: Isomorphism

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. $𝑇$ is an isomorphism iff $𝑇$ is invertible. If there exists an isomorphism between $𝑉$ and $𝑊$ , we say that $𝑉$ and $𝑊$ are isomorphic, denoted $𝑉 ≅ 𝑊$ .

Proposition 9: Isomorphisms are an Equivalence Relation

Let $𝑋$ , $𝑌$ , and $𝑍$ be vector spaces over a field $𝐹$ . Then the following hold:

Reflexivity. $𝑋 ≅ 𝑋$ .
Symmetry. If $𝑋 ≅ 𝑌$ , then $𝑌 ≅ 𝑋$ .
Transitivity. If $𝑋 ≅ 𝑌$ and $𝑌 ≅ 𝑍$ , then $𝑋 ≅ 𝑍$ .

Proof: Proposition 9.

Let $𝑋$ , $𝑌$ , and $𝑍$ be vector spaces over a field $𝐹$ .

Reflexivity. The identity map ${id}_{𝑋} : 𝑋 \to 𝑋$ is an isomorphism, so $𝑋 ≅ 𝑋$ .
Symmetry. If $𝑇 : 𝑋 \to 𝑌$ is an isomorphism, then the inverse map $𝑇^{- 1} : 𝑌 \to 𝑋$ is also an isomorphism, so $𝑌 ≅ 𝑋$ .
Transitivity. If $𝑇 : 𝑋 \to 𝑌$ and $𝑆 : 𝑌 \to 𝑍$ are isomorphisms, then the composition $𝑆 \circ 𝑇 : 𝑋 \to 𝑍$ is also an isomorphism ( ${(𝑆 \circ 𝑇)}^{- 1} = 𝑇^{- 1} \circ 𝑆^{- 1}$ ), so $𝑋 ≅ 𝑍$ .

Theorem 45: Classification of Finite-Dimensional Vector Spaces

Let $𝑉$ and $𝑊$ be finite-dimensional vector spaces over a field $𝐹$ . Then $𝑉 ≅ 𝑊$ if and only if $\dim (𝑉) = \dim (𝑊)$ .

Corollary 24: Isomorphisms to Coordinate Spaces

Let $𝑉$ be a finite $𝑛$ -dimensional vector space over a field $𝐹$ . Then $𝑉 ≅ 𝐹^{𝑛}$ .

Specifically, we can choose any basis $ℬ$ for $𝑉$ and use the coordinate map $𝐯 \mapsto {[𝐯]}_{ℬ}$ to construct an isomorphism from $𝑉$ to $𝐹^{𝑛}$ .

Corollary 25: Matrices are Isomorphic to Coordinate Vectors

$𝐹^{𝑚 \times 𝑛} ≅ 𝐹^{𝑚 𝑛} .$

Example.

Let $𝐴 = (\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}) \in ℝ^{2 \times 3}$ . Then $𝐴$ is isomorphic to the coordinate vector $(1, 2, 3, 4, 5, 6) \in ℝ^{6}$ under the isomorphism $𝐴 \mapsto {[𝐴]}_{ℬ}$ , where:

ℬ = {(\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}), (\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}), (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), (\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}), (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix})} .

Change of Basis and Similarity

Recall from Theorem 35 that for any linear map $𝑇 : 𝑉 \to 𝑊$ and any bases $ℬ$ and $𝒞$ for $𝑉$ and $𝑊$ , respectively, we have:

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

Taking $𝑇$ to be the identity map ${id}_{𝑉} : 𝑉 \to 𝑉$ , and choosing two, potentially different bases $ℬ$ and $𝒞$ for $𝑉$ , we have:

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

Corollary 26

Let $𝑉$ be a vector space and choose bases $ℬ$ and $𝒞$ for $𝑉$ . Then for any vector $𝐯 \in 𝑉$ :

{[𝐯]}_{𝒞} = {[{id}_{𝑉}]}_{𝒞 ℬ} {[𝐯]}_{ℬ} .

Suppose $𝑉$ is an $𝑛$ -dimensional vector space. Then ${[{id}_{𝑉}]}_{𝒞 ℬ}$ is an $𝑛 \times 𝑛$ matrix, which when applied to the coordinate vector ${[𝐯]}_{ℬ} \in 𝐹^{𝑛}$ , produces the coordinate vector ${[𝐯]}_{𝒞} \in 𝐹^{𝑛}$ .

Definition 49: Change of Basis Matrix

Let $𝑉$ be an $𝑛$ -dimensional vector space over a field $𝐹$ , and let $ℬ$ and $𝒞$ be bases for $𝑉$ . The change of basis matrix from the basis $ℬ$ to the basis $𝒞$ , denoted $𝑃_{𝒞 ℬ}$ , is given by:

𝑃_{𝒞 ℬ} = {[{id}_{𝑉}]}_{𝒞 ℬ} = (\begin{matrix} | & | \\ {[𝐛_{1}]}_{𝒞} & \dots & {[𝐛_{𝑛}]}_{𝒞} \\ | & | \end{matrix}),

where $𝐛_{1}, 𝐛_{2}, \dots, 𝐛_{𝑛}$ are the vectors in the basis $ℬ$ . It holds that $𝑃_{𝒞 ℬ} \in 𝐹^{𝑛 \times 𝑛}$ .

Theorem 46: Invertibility of Change of Basis Matrix

Let $𝑉$ be an $𝑛$ -dimensional vector space over a field $𝐹$ , and let $ℬ$ and $𝒞$ be bases for $𝑉$ . Then the change of basis matrix $𝑃_{𝒞 ℬ} \in 𝐹^{𝑛 \times 𝑛}$ is invertible, with:

𝑃_{𝒞 ℬ}^{- 1} = 𝑃_{ℬ 𝒞} .

Definition 50: Change of Basis Transformation

Let $𝑉$ be an $𝑛$ -dimensional vector space over a field $𝐹$ , and let $ℬ$ and $𝒞$ be bases for $𝑉$ . The change of basis transformation from the basis $ℬ$ to the basis $𝒞$ , denoted $𝑇_{𝒞 ℬ}$ , is the linear map $𝑇_{𝒞 ℬ} : 𝐹^{𝑛} \to 𝐹^{𝑛}$ defined such that:

𝑇_{𝒞 ℬ} ({[𝐯]}_{ℬ}) = {[𝐯]}_{𝒞} .

Corollary 27

The change of basis transformation $𝑇_{𝒞 ℬ}$ is concretely defined by:

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

Theorem 47: Change of Basis for Matrices

Let $𝑇 : 𝑉 \to 𝑉$ be a linear map on an $𝑛$ -dimensional vector space $𝑉$ over a field $𝐹$ , and let $ℬ$ and $𝒞$ be bases for $𝑉$ . Then the standard matrices of $𝑇$ with respect to the bases $ℬ$ and $𝒞$ are related by:

{[𝑇]}_{𝒞 𝒞} = 𝑃_{𝒞 ℬ} {[𝑇]}_{ℬ ℬ} 𝑃_{ℬ 𝒞} .

Definition 51: Similarity of Matrices

Let $𝐴, 𝐵 \in 𝐹^{𝑛 \times 𝑛}$ be square matrices. $𝐴$ and $𝐵$ are similar iff there exists an invertible matrix $𝑃 \in 𝐹^{𝑛 \times 𝑛}$ such that:

𝐵 = 𝑃^{- 1} 𝐴 𝑃 .

Corollary 28

Two matrices $𝐴$ and $𝐵$ are similar if and only if there exists a linear map $𝑇 : 𝑉 \to 𝑉$ and bases $ℬ$ and $𝒞$ for $𝑉$ such that $𝐴 = {[𝑇]}_{ℬ ℬ}$ and $𝐵 = {[𝑇]}_{𝒞 𝒞}$ .

That is, two matrices are similar if and only if they represent the same linear map, but with respect to potentially different bases.

Exercises

Exercise 66.

Suppose

𝑇 : 𝑉 \to 𝑊

is an isomorphism between

𝑛

-dimensional vector spaces. Show that if

{𝐛_{1}, 𝐛_{2}, \dots, 𝐛_{𝑛}}

is a basis for

𝑉

, then

{𝑇 (𝐛_{1}), 𝑇 (𝐛_{2}), \dots, 𝑇 (𝐛_{𝑛})}

is a basis for

𝑊

Exercise 67.

Let the entries of a

2 \times 2

real-valued matrix

𝐴

be chosen uniformly at random from the set

{- 2, 1, 0, 1, 2}

. Find the probability that

𝐴

is invertible.

Exercise 68.

Let

𝐴 \in 𝐹^{𝑛 \times 𝑛}

. Show that if

𝐴^{2} = 0

, then

𝐴

is not invertible.

Exercise 69.

Suppose

𝐴, 𝐵 \in 𝐹^{𝑛 \times 𝑛}

with

𝐵 \neq 0

. Prove or disprove that if

𝐴 𝐵 = 0

, then

𝐴

is not invertible.

Exercise 70.

Let

ℙ_{2} (ℝ)

be the vector space of all polynomials of degree at most

2

over the field

ℝ

. Let

ℬ = {1, 𝑥, 𝑥^{2}}

and

𝒞 = {1, 𝑥 - 1, {(𝑥 - 1)}^{2}}

be bases for

ℙ_{2} (ℝ)

. Find the change of basis matrix

𝑃_{𝒞 ℬ}