Introduction to Linear Maps

Linear Maps

Definition 26: Linear Map

Let $𝑉$ and $𝑊$ be vector spaces over the same field $𝐹$ . A linear map (or linear transformation) from $𝑉$ to $𝑊$ is a function $𝑇 : 𝑉 \to 𝑊$ such that the following properties hold:

Additivity. $\forall 𝐮, 𝐯 \in 𝑉, 𝑇 (𝐮 + 𝐯) = 𝑇 (𝐮) + 𝑇 (𝐯)$ .
Homogeneity. $\forall 𝐯 \in 𝑉, \forall 𝑐 \in 𝐹, 𝑇 (𝑐 𝐯) = 𝑐 𝑇 (𝐯)$ .

These two properties are known as the linearity conditions or linear map axioms.

Terminology: Domain and Codomain

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. The domain of $𝑇$ is $𝑉$ , and the codomain of $𝑇$ is $𝑊$ .

Terminology: Preimage and Image of a Vector under a Linear Map

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map such that $𝑇 (𝐯) = 𝐰$ for some $𝐯 \in 𝑉$ and $𝐰 \in 𝑊$ .

$𝐯$ is called the preimage of $𝐰$ under $𝑇$ .
$𝐰$ is called the image of $𝐯$ under $𝑇$ .

In which case, we say that $𝑇$ maps $𝐯$ to $𝐰$ .

Notation

The shorthand notation $𝐯 \mapsto 𝑇 (𝐯)$ is often used to denote the action of a linear map $𝑇$ on a vector $𝐯$ . For example, if $𝑇 (𝐯) = 𝐰$ , we would write $𝐯 \mapsto 𝐰$ to indicate that $𝑇$ maps $𝐯$ to $𝐰$ .

Example: Differentiation is a linear map.

Let $ℙ_{𝑛} (ℝ)$ denote the set of polynomials with real coefficients of degree at most $𝑛$ . Define a function $𝑇 : ℙ_{𝑛} (ℝ) \to ℙ_{𝑛 - 1} (ℝ)$ by $𝑓 \mapsto 𝑓^{'}$ , where $𝑓^{'}$ is the derivative of $𝑓$ . Show that $𝑇$ is a linear map.

Proof.

To show that $𝑇$ is a linear map, we need to verify the two linearity conditions: additivity and homogeneity.

Additivity. Let $𝑓, 𝑔 \in ℙ_{𝑛} (ℝ)$ . We need to show that $𝑇 (𝑓 + 𝑔) = 𝑇 (𝑓) + 𝑇 (𝑔)$ . We have:
$𝑇 (𝑓 + 𝑔) = {(𝑓 + 𝑔)}^{'} = 𝑓^{'} + 𝑔^{'} = 𝑇 (𝑓) + 𝑇 (𝑔) .$
Thus, additivity holds.
Homogeneity. Let $𝑓 \in ℙ_{𝑛} (ℝ)$ and let $𝑐 \in ℝ$ . We need to show that $𝑇 (𝑐 𝑓) = 𝑐 𝑇 (𝑓)$ . We have:
$𝑇 (𝑐 𝑓) = {(𝑐 𝑓)}^{'} = 𝑐 𝑓^{'} = 𝑐 𝑇 (𝑓) .$
Thus, homogeneity holds.

Since both linearity conditions are satisfied, we conclude that $𝑇$ is a linear map. ∎

Notation: Set of Linear Maps

The set of all linear maps from $𝑉$ to $𝑊$ may be denoted by $ℒ (𝑉, 𝑊)$ or $ℒ_{𝑉 \to 𝑊}$ . Additionally, the set of all linear maps from $𝑉$ to itself may be denoted by $ℒ (𝑉)$ or $ℒ_{𝑉}$ .

Theorem 26: Alternative Characterization of Linearity

Let $𝑉$ and $𝑊$ be vector spaces over the same field $𝐹$ , and let $𝑇 : 𝑉 \to 𝑊$ be a function. Then $𝑇$ is a linear map if and only if:

\forall 𝐮, 𝐯 \in 𝑉, \forall 𝑎, 𝑏 \in 𝐹, 𝑇 (𝑎 𝐮 + 𝑏 𝐯) = 𝑎 𝑇 (𝐮) + 𝑏 𝑇 (𝐯) .

Corollary 13: General Linearity

Let $𝑉$ and $𝑊$ be vector spaces over the same field $𝐹$ , and let $𝑇 : 𝑉 \to 𝑊$ be a function. Then $𝑇$ is a linear map if and only if for all $𝑐_{1}, 𝑐_{2}, \dots, 𝑐_{𝑘} \in 𝐹$ and all $𝐯_{1}, 𝐯_{2}, \dots, 𝐯_{𝑘} \in 𝑉$ :

𝑇 (\sum_{𝑖 = 1}^{𝑘} 𝑐_{𝑖} 𝐯_{𝑖}) = \sum_{𝑖 = 1}^{𝑘} 𝑐_{𝑖} 𝑇 (𝐯_{𝑖}) .

Theorem 27: Zero Vector Preservation of Linear Maps

Let $𝑉$ and $𝑊$ be vector spaces over the same field $𝐹$ , and let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Then $𝑇$ preserves the zero vector, i.e. $𝑇 (𝟎_{𝑉}) = 𝟎_{𝑊}$ .

Proof: Theorem 27.

Assume

𝑇 : 𝑉 \to 𝑊

over

𝐹

is a linear map that satisfies the linearity conditions in Definition 26. Then by Theorem 3,

𝑇 (𝟎_{𝑉}) = 𝑇 (0_{𝐹} 𝐯)

for any

𝐯 \in 𝑉

. By homogeneity,

𝑇 (0_{𝐹} 𝐯) = 0_{𝐹} 𝑇 (𝐯)

. Since

𝑇 (𝐯) \in 𝑊

, by Theorem 3 again,

0_{𝐹} 𝑇 (𝐯) = 𝟎_{𝑊}

. Thus, by transitivity of equality,

𝑇 (𝟎_{𝑉}) = 𝟎_{𝑊}

. ∎

We can apply multiple linear maps in succession by composing them, and the result is still a linear map. This allows us to build more complex linear maps from simpler ones, and to analyze the structure of linear maps in terms of their compositions.

Theorem 28: Composition of Linear Maps

Define the linear maps $𝑈 : 𝑉 \to 𝑊$ and $𝑇 : 𝑊 \to 𝑋$ . Then the function composition of $𝑇$ and $𝑈$ , denoted $𝑇 \circ 𝑈$ , is a linear map from $𝑉$ to $𝑋$ , where $𝑇 \circ 𝑈$ is defined by:

(𝑇 \circ 𝑈) (𝐯) = 𝑇 (𝑈 (𝐯)) .

Notation

The composition $𝑇 \circ 𝑈$ of two linear maps may be denoted by juxtaposition, i.e. $𝑇 𝑈$ . This should not be confused with the product of two functions. Furthermore, the repeated composition of a linear map $𝑛$ times may be denoted $𝑇^{𝑛}$ , and should not be confused with the $𝑛$ ^th power of a function:

𝑇 𝑈 = 𝑇 \circ 𝑈, 𝑇^{𝑛} = \underset{𝑛 times}{\underset{⏟}{𝑇 \circ 𝑇 \circ \dots \circ 𝑇}} .

Proof: Theorem 28.

Let $𝑈 : 𝑉 \to 𝑊$ and $𝑇 : 𝑊 \to 𝑋$ be linear maps. We need to show that $𝑇 \circ 𝑈 : 𝑉 \to 𝑋$ is a linear map, i.e. it satisfies the linearity conditions.

Additivity. Let $𝐮, 𝐯 \in 𝑉$ . We need to show that $(𝑇 \circ 𝑈) (𝐮 + 𝐯) = (𝑇 \circ 𝑈) (𝐮) + (𝑇 \circ 𝑈) (𝐯)$ . We have:
$\begin{matrix} (𝑇 \circ 𝑈) (𝐮 + 𝐯) = 𝑇 (𝑈 (𝐮 + 𝐯)) = 𝑇 (𝑈 (𝐮) + 𝑈 (𝐯)) & = 𝑇 (𝑈 (𝐮)) + 𝑇 (𝑈 (𝐯)) \\ = (𝑇 \circ 𝑈) (𝐮) + (𝑇 \circ 𝑈) (𝐯) . \end{matrix}$
Thus, additivity holds.
Homogeneity. Let $𝐯 \in 𝑉$ and let $𝑐 \in 𝐹$ . We need to show that $(𝑇 \circ 𝑈) (𝑐 𝐯) = 𝑐 (𝑇 \circ 𝑈) (𝐯)$ . We have:
$(𝑇 \circ 𝑈) (𝑐 𝐯) = 𝑇 (𝑈 (𝑐 𝐯)) = 𝑇 (𝑐 𝑈 (𝐯)) = 𝑐 𝑇 (𝑈 (𝐯)) = 𝑐 (𝑇 \circ 𝑈) (𝐯) .$
Thus, homogeneity holds.

Since both linearity conditions are satisfied, we conclude that $𝑇 \circ 𝑈$ is a linear map. ∎

The simplest linear maps are the identity and zero linear maps. The identity linear map on a vector space $𝑉$ maps every vector to itself:

Definition 27: Identity Linear Map

Let $𝑉$ be a vector space. The identity map or identity transformation on $𝑉$ , denoted ${id}_{𝑉}$ , is the linear map from $𝑉$ to itself defined by:

{id}_{𝑉} (𝐯) = 𝐯 for all 𝐯 \in 𝑉 .

That is, ${id}_{𝑉} : 𝑉 \to 𝑉$ is the linear map defined by $𝐯 \mapsto 𝐯$ .

Corollary 14

For any linear map $𝑇 : 𝑉 \to 𝑊$ , ${id}_{𝑊} \circ 𝑇 = 𝑇 \circ {id}_{𝑉} = 𝑇$ .

Definition 28: Zero Linear Map

Let $𝑉$ and $𝑊$ be vector spaces over the same field $𝐹$ . The zero map or zero transformation from $𝑉$ to $𝑊$ , denoted $0_{𝑉 \to 𝑊}$ , is the linear map defined by:

0_{𝑉 \to 𝑊} (𝐯) = 𝟎_{𝑊} for all 𝐯 \in 𝑉 .

That is, $0_{𝑉 \to 𝑊} : 𝑉 \to 𝑊$ is the linear map defined by $𝐯 \mapsto 𝟎_{𝑊}$ .

Theorem 29: Linearity of the Basis Transformation

Let $𝑉$ be an $𝑛$ -dimensional vector space over a field $𝐹$ , and suppose $ℬ$ is a basis for $𝑉$ . Then the basis transformation ${[\cdot]}_{ℬ} : 𝑉 \to 𝐹^{𝑛}$ defined by $𝐯 \mapsto {[𝐯]}_{ℬ}$ is a linear map.

Proof: Theorem 29.

Let $𝑉$ be an $𝑛$ -dimensional vector space over a field $𝐹$ , and suppose $ℬ = {𝐛_{1}, \dots, 𝐛_{𝑛}}$ is an ordered basis for $𝑉$ . Let $𝐮, 𝐯 \in 𝑉$ and let $𝑐 \in 𝐹$ , and suppose ${[𝐮]}_{ℬ} = (𝛼_{1}, \dots, 𝛼_{𝑛})$ and ${[𝐯]}_{ℬ} = (𝛽_{1}, \dots, 𝛽_{𝑛})$ . Then we can write:

𝐮 = \sum_{𝑖 = 1}^{𝑛} 𝛼_{𝑖} 𝐛_{𝑖}, 𝐯 = \sum_{𝑖 = 1}^{𝑛} 𝛽_{𝑖} 𝐛_{𝑖} .

We need to show that the basis transformation ${[\cdot]}_{ℬ} : 𝑉 \to 𝐹^{𝑛}$ defined by $𝐯 \mapsto {[𝐯]}_{ℬ}$ satisfies linearity conditions.

Additivity. We have:
$\begin{matrix} {[𝐮 + 𝐯]}_{ℬ} & = {[\sum_{𝑖 = 1}^{𝑛} 𝛼_{𝑖} 𝐛_{𝑖} + \sum_{𝑖 = 1}^{𝑛} 𝛽_{𝑖} 𝐛_{𝑖}]}_{ℬ} & by substitution \\ = {[\sum_{𝑖 = 1}^{𝑛} (𝛼_{𝑖} + 𝛽_{𝑖}) 𝐛_{𝑖}]}_{ℬ} & by distributivity \\ = (𝛼_{1} + 𝛽_{1}, \dots, 𝛼_{𝑛} + 𝛽_{𝑛}) & by definition of basis coordinates \\ = (𝛼_{1}, \dots, 𝛼_{𝑛}) + (𝛽_{1}, \dots, 𝛽_{𝑛}) & by vector addition in 𝐹^{𝑛} \\ = {[𝐮]}_{ℬ} + {[𝐯]}_{ℬ} & by substitution . \end{matrix}$
Homogeneity. We have:
$\begin{matrix} {[𝑐 𝐮]}_{ℬ} & = {[𝑐 \sum_{𝑖 = 1}^{𝑛} 𝛼_{𝑖} 𝐛_{𝑖}]}_{ℬ} & by substitution \\ = {[\sum_{𝑖 = 1}^{𝑛} (𝑐 𝛼_{𝑖}) 𝐛_{𝑖}]}_{ℬ} & by distributivity \\ = (𝑐 𝛼_{1}, \dots, 𝑐 𝛼_{𝑛}) & by definition of basis coordinates \\ = 𝑐 (𝛼_{1}, \dots, 𝛼_{𝑛}) & by scalar multiplication in 𝐹^{𝑛} \\ = 𝑐 {[𝐮]}_{ℬ} & by substitution . \end{matrix}$

Since both linearity conditions are satisfied, ${[\cdot]}_{ℬ}$ is a linear map. ∎

Theorem 30: Sets of Linear Maps are Vector Spaces

Let $𝑉$ and $𝑊$ be vector spaces over the same field $𝐹$ . Then the set of all linear maps from $𝑉$ to $𝑊$ , denoted $ℒ (𝑉, 𝑊)$ , is a vector space over $𝐹$ under the following operations:

Pointwise addition. For $𝑇, 𝑆 \in ℒ (𝑉, 𝑊)$ , define $(𝑇 + 𝑆) (𝐯) = 𝑇 (𝐯) + 𝑆 (𝐯)$ for all $𝐯 \in 𝑉$ .
Scalar multiplication. For $𝑇 \in ℒ (𝑉, 𝑊)$ and $𝑐 \in 𝐹$ , define $(𝑐 𝑇) (𝐯) = 𝑐 𝑇 (𝐯)$ for all $𝐯 \in 𝑉$ .

Furthermore, $\dim (ℒ (𝑉, 𝑊)) = \dim (𝑉) \dim (𝑊)$ .

Kernel, Nullity, and Injectivity

Definition 29: Kernel of a Linear Map

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. The kernel or null space of $𝑇$ , denoted $\ker (𝑇)$ , is the set of all vectors in $𝑉$ that are mapped to the zero vector in $𝑊$ . Formally:

\ker (𝑇) = {𝐯 \in 𝑉 | 𝑇 (𝐯) = 𝟎_{𝑊}} .

Theorem 31: Kernel is a Subspace of the Domain

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Then $\ker (𝑇)$ is a subspace of $𝑉$ .

Proof: Kernel is a Subspace of the Domain.

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Checking the Subspace Criteria:

Existence of vector additive identity. Since $𝑇 (𝟎_{𝑉}) = 𝟎_{𝑊}$ by Theorem 27, $𝟎_{𝑉} \in \ker (𝑇)$ .
Closure under vector addition. Let $𝐮, 𝐯 \in \ker (𝑇)$ . Then $𝑇 (𝐮) = 𝑇 (𝐯) = 𝟎_{𝑊}$ . By additivity, $𝑇 (𝐮 + 𝐯) = 𝑇 (𝐮) + 𝑇 (𝐯) = 𝟎_{𝑊} + 𝟎_{𝑊} = 𝟎_{𝑊}$ , so $𝐮 + 𝐯 \in \ker (𝑇)$ .
Closure under scalar multiplication. Let $𝐯 \in \ker (𝑇)$ and let $𝑐 \in 𝐹$ . Then $𝑇 (𝐯) = 𝟎_{𝑊}$ . By homogeneity, $𝑇 (𝑐 𝐯) = 𝑐 𝑇 (𝐯) = 𝑐 𝟎_{𝑊} = 𝟎_{𝑊}$ , so $𝑐 𝐯 \in \ker (𝑇)$ . ∎

Definition 30: Nullity of a Linear Map

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. The nullity of $𝑇$ is the dimension of the kernel of $𝑇$ :

nullity (𝑇) = \dim (\ker (𝑇)) .

Theorem 32: Nullity Theorem

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Then $𝑇$ is injective if and only if $nullity (𝑇) = 0$ ; that is, $𝑇$ is injective if and only if $\ker (𝑇) = {𝟎_{𝑉}}$ .

Image, Rank, and Surjectivity

Definition 31: Image of a Linear Map

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. The image or range of $𝑇$ , denoted $im (𝑇)$ , is the set of all vectors in $𝑊$ that are the image of some vector in $𝑉$ under $𝑇$ . Formally:

im (𝑇) = {𝐰 \in 𝑊 | \exists 𝐯 \in 𝑉 s.t. 𝑇 (𝐯) = 𝐰} .

Theorem 33: Image is a Subspace of the Codomain

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Then $im (𝑇)$ is a subspace of $𝑊$ .

Proof: Image is a Subspace of the Codomain.

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. Checking the Subspace Criteria:

Existence of vector additive identity. Since $𝑇 (𝟎_{𝑉}) = 𝟎_{𝑊}$ by Theorem 27, $𝟎_{𝑊} \in im (𝑇)$ .
Closure under vector addition. Let $𝐰_{1}, 𝐰_{2} \in im (𝑇)$ . Then there exist $𝐯_{1}, 𝐯_{2} \in 𝑉$ such that $𝑇 (𝐯_{1}) = 𝐰_{1}$ and $𝑇 (𝐯_{2}) = 𝐰_{2}$ . By additivity, $𝑇 (𝐯_{1} + 𝐯_{2}) = 𝑇 (𝐯_{1}) + 𝑇 (𝐯_{2}) = 𝐰_{1} + 𝐰_{2}$ . Since $𝐯_{1} + 𝐯_{2} \in 𝑉$ , it is the preimage of $𝐰_{1} + 𝐰_{2}$ , so $𝐰_{1} + 𝐰_{2} \in im (𝑇)$ .
Closure under scalar multiplication. Let $𝐰 \in im (𝑇)$ and let $𝑐 \in 𝐹$ . Then there exists $𝐯 \in 𝑉$ such that $𝑇 (𝐯) = 𝐰$ . By homogeneity, $𝑇 (𝑐 𝐯) = 𝑐 𝑇 (𝐯) = 𝑐 𝐰$ , so $𝑐 𝐰 \in im (𝑇)$ . ∎

Notation

Let $𝑓 : 𝑈 \to 𝑉$ be a function. Then for any subset $𝑆 \subseteq 𝑈$ , $𝑓 (𝑆)$ (called the image of $𝑆$ under $𝑓$ ) denotes the set of all images of elements in $𝑆$ under $𝑓$ :

𝑓 (𝑆) = {𝑓 (𝑠) \in 𝑉 | 𝑠 \in 𝑆} \subseteq im (𝑓) .

Definition 32: Rank of a Linear Map

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. The rank of $𝑇$ is the dimension of the image of $𝑇$ :

rank (𝑇) = \dim (im (𝑇)) .

Theorem 34: Rank-Nullity Theorem

Let $𝑇 : 𝑉 \to 𝑊$ be a linear map. If $𝑉$ is finite-dimensional, then $𝑊$ is also finite-dimensional, and the following equation holds:

rank (𝑇) + nullity (𝑇) = \dim (𝑉) .

Terminology

A transformation $𝑇$ is bijective (sometimes also called a one-to-one correspondence) if $𝑇$ is both injective and surjective.

Exercises

Exercise 51.

Prove the Alternative Characterization of Linearity.

Exercise 52.

Show that the transformation

𝑇 : ℝ^{2} \to ℝ

defined by

𝑇 (𝑥, 𝑦) = 𝑥 𝑦

is not linear.

Exercise 53.

Let $𝑇 : 𝐹^{\infty} \to 𝐹^{\infty}$ be the transformation defined by:

𝑇 (𝑎_{1}, 𝑎_{2}, 𝑎_{3}, \dots) = (0, 𝑎_{1}, 𝑎_{2}, 𝑎_{3}, \dots) for all 𝑎_{1}, 𝑎_{2}, 𝑎_{3}, \dots \in 𝐹 .

Determine, with proof, whether $𝑇$ is a linear map.

Exercise 54.

Suppose

𝑇

is a linear map. Show that if the set

{𝐯_{1}, 𝐯_{2}, \dots, 𝐯_{𝑘}}

is linearly independent, then the set

{𝑇 (𝐯_{1}), 𝑇 (𝐯_{2}), \dots, 𝑇 (𝐯_{𝑘})}

is also linearly independent.

Exercise 55.

Let

𝑉

and

𝑊

be finite-dimensional vector spaces such that

\dim (𝑉) > \dim (𝑊)

. Show that there does not exist an injective linear map

𝑇 \in ℒ (𝑉, 𝑊)

Exercise 56.

Let $𝑇, 𝑆 \in ℒ (𝑉, 𝑊)$ be linear maps. Show that:

\ker (𝑇) \cap \ker (𝑆) \subseteq \ker (𝑇 + 𝑆) .

Exercise 57.

For linear maps $𝑇 : 𝑈 \to 𝑉$ and $𝑆 : 𝑉 \to 𝑊$ , both with finite rank, show that:

rank (𝑆 \circ 𝑇) \leq \min (rank (𝑆), rank (𝑇)) .

Exercise 58.

Suppose

𝑉

is a finite-dimensional vector space. Show that there exists a linear map

𝑇 : 𝑉 \to 𝑉

such that

\ker (𝑇) = im (𝑇)

if and only if

\dim (𝑉)

is even.

Exercise 59.

A linear map

𝑇

is called idempotent iff

𝑇^{2} = 𝑇

. Suppose that

𝑇 : 𝑉 \to 𝑉

is a linear map, where

𝑉

is finite-dimensional. Show that if

𝑇

is idempotent, then

𝑉 = \ker (𝑇) \oplus im (𝑇)

Exercise 60.

Suppose $𝑉$ is a finite-dimensional vector space, and $𝑇 \in ℒ (𝑉)$ . Show that:

$\ker (𝑇) \subseteq \ker (𝑇^{2}) \subseteq \ker (𝑇^{3}) \subseteq \dots$
If for some $𝑘$ , $\ker (𝑇^{𝑘}) = \ker (𝑇^{𝑘 + 1})$ , then $\ker (𝑇^{𝑘}) = \ker (𝑇^{𝑚})$ for all $𝑚 \geq 𝑘$

Exercise 61.

Let $𝑉$ and $𝑊$ be finite-dimensional vector spaces. The set of all linear maps from $𝑉$ to $𝑊$ , $ℒ (𝑉, 𝑊)$ , is itself a vector space under pointwise addition and scalar multiplication.

Fix some vector $𝐯_{0} \in 𝑉$ to define the following function $Φ : ℒ (𝑉, 𝑊) \to 𝑊$ :

Φ (𝑇) = 𝑇 (𝐯_{0}) .

Show that $Φ$ is a linear map.
Find $nullity (Φ)$ and $rank (Φ)$ in terms of $\dim (𝑉)$ and $\dim (𝑊)$ .
Find $\dim (ℒ (𝑉, 𝑊))$ in terms of $\dim (𝑉)$ and $\dim (𝑊)$ .