1. Linear Maps

Definition 4.1: Linear Map

Let $V$ and $W$ be vector spaces over the same field $F$ . A function $T: V \to W$ is called a linear map (or linear transformation) if for all vectors $u, v \in V$ and all scalars $\alpha, \beta \in F$ :

T(\alpha u + \beta v) = \alpha T(u) + \beta T(v)

Remark 4.1: Equivalent Formulation

The linearity condition can be split into two separate properties:

Additivity: $T(u + v) = T(u) + T(v)$ for all $u, v \in V$
Homogeneity: $T(\alpha v) = \alpha T(v)$ for all $\alpha \in F$ and $v \in V$

Theorem 4.1: Basic Properties of Linear Maps

Let $T: V \to W$ be a linear map. Then:

$T(0_V) = 0_W$ (T sends the zero vector to the zero vector)
$T(-v) = -T(v)$ for all $v \in V$
$T(v - w) = T(v) - T(w)$ for all $v, w \in V$

Proof of Theorem 4.1 (Property 1):

Using homogeneity with $\alpha = 0$ :

T(0_V) = T(0 \cdot v) = 0 \cdot T(v) = 0_W

∎

Example 4.1: Standard Examples of Linear Maps

Zero Map: $0: V \to W$ defined by $0(v) = 0_W$ for all $v \in V$
Identity Map: $I_V: V \to V$ defined by $I_V(v) = v$ for all $v \in V$
Differentiation: $D: P_n(\mathbb{R}) \to P_{n-1}(\mathbb{R})$ defined by $D(p) = p'$

Example 4.2: Non-Examples

The following are NOT linear maps:

Translation: $T(x) = x + c$ for $c \neq 0$ . Since $T(0) = c \neq 0$ .
Product map: $T(x_1, x_2) = (x_1 x_2, x_1 + x_2)$ . Not additive.

Example 4.3: More Linear Map Examples

Projection: $\pi: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $\pi(x, y, z) = (x, y)$
Rotation: In $\mathbb{R}^2$ , rotation by angle $\theta$ is linear
Integration: $I: P_n(\mathbb{R}) \to P_{n+1}(\mathbb{R})$ defined by $I(p)(x) = \int_0^x p(t) dt$
Matrix multiplication: For fixed $A \in M_{m \times n}(F)$ , the map $T: F^n \to F^m$ defined by $T(x) = Ax$ is linear

Theorem 1.1: Linear Maps Preserve Linear Combinations

A map $T: V \to W$ is linear if and only if for all $v_1, \ldots, v_k \in V$ and $\alpha_1, \ldots, \alpha_k \in F$ :

T\left(\sum_{i=1}^k \alpha_i v_i\right) = \sum_{i=1}^k \alpha_i T(v_i)

2. Kernel (Null Space)

Definition 4.2: Kernel

The kernel (or null space) of a linear map $T: V \to W$ is:

\ker(T) = \{v \in V : T(v) = 0_W\}

Theorem 4.2: Kernel is a Subspace

For any linear map $T: V \to W$ , $\ker(T)$ is a subspace of $V$ .

Proof:

We verify the subspace criterion:

Contains zero: $T(0) = 0$ , so $0 \in \ker(T)$
Closed under addition: If $u, v \in \ker(T)$ , then $T(u + v) = T(u) + T(v) = 0 + 0 = 0$
Closed under scalar multiplication: If $v \in \ker(T)$ and $\alpha \in F$ , then $T(\alpha v) = \alpha T(v) = \alpha \cdot 0 = 0$

∎

Theorem 4.3: Injectivity via Kernel

A linear map $T: V \to W$ is injective (one-to-one) if and only if $\ker(T) = \{0\}$ .

Proof:

(⇒) If $T$ is injective and $v \in \ker(T)$ , then $T(v) = 0 = T(0)$ . By injectivity, $v = 0$ .

(⇐) If $\ker(T) = \{0\}$ and $T(u) = T(v)$ , then:

T(u - v) = T(u) - T(v) = 0

So $u - v \in \ker(T) = \{0\}$ , hence $u = v$ .

∎

Example 4.3: Kernel Examples

For differentiation $D: P_n \to P_{n-1}$ , $\ker(D) = \{c : c \in \mathbb{R}\}$ (constant polynomials)
For $T(x, y) = (x + y, x + y)$ , $\ker(T) = \{(t, -t) : t \in \mathbb{R}\}$
For the identity map, $\ker(I_V) = \{0\}$

Theorem 2.1: Kernel and Linear Independence

If $\{v_1, \ldots, v_k\}$ is linearly independent and $T: V \to W$ is injective, then $\{T(v_1), \ldots, T(v_k)\}$ is linearly independent.

Proof:

If $\sum \alpha_i T(v_i) = 0$ , then $T(\sum \alpha_i v_i) = 0$ . Since $\ker(T) = \{0\}$ (injectivity), $\sum \alpha_i v_i = 0$ . By independence, all $\alpha_i = 0$ .

∎

Remark 2.1: Kernel as Measure of Non-Injectivity

The kernel measures how much a linear map "collapses" the domain. A large kernel means many different vectors map to the same output, indicating non-injectivity.

3. Image (Range)

Definition 4.3: Image

The image (or range) of a linear map $T: V \to W$ is:

\text{im}(T) = \{T(v) : v \in V\}

Theorem 4.4: Image is a Subspace

For any linear map $T: V \to W$ , $\text{im}(T)$ is a subspace of $W$ .

Proof:

We verify the subspace criterion:

Contains zero: $T(0) = 0$ , so $0 \in \text{im}(T)$
Closed under addition: If $w_1, w_2 \in \text{im}(T)$ , say $w_i = T(v_i)$ , then $w_1 + w_2 = T(v_1) + T(v_2) = T(v_1 + v_2) \in \text{im}(T)$
Closed under scalar multiplication: If $w = T(v) \in \text{im}(T)$ and $\alpha \in F$ , then $\alpha w = \alpha T(v) = T(\alpha v) \in \text{im}(T)$

∎

Theorem 4.5: Surjectivity via Image

A linear map $T: V \to W$ is surjective (onto) if and only if $\text{im}(T) = W$ .

Example 4.4: Image Examples

For $T(x, y, z) = (x, y, 0)$ , $\text{im}(T) = \{(x, y, 0) : x, y \in \mathbb{R}\}$ (the xy-plane)
For the identity map, $\text{im}(I_V) = V$
For the zero map, $\text{im}(0) = \{0\}$

Theorem 3.1: Image and Spanning Sets

If $\{v_1, \ldots, v_k\}$ spans $V$ , then $\{T(v_1), \ldots, T(v_k)\}$ spans $\text{im}(T)$ . If $T$ is surjective, then $\{T(v_1), \ldots, T(v_k)\}$ spans $W$ .

Remark 3.1: Image as Measure of Surjectivity

The image measures how much of the codomain is "covered" by the map. If $\text{im}(T) = W$ , the map is surjective; otherwise, it misses some vectors in $W$ .

4. Computing Kernel and Image

Theorem 4.6: Image from Basis

If $\{v_1, \ldots, v_n\}$ spans $V$ , then $\{T(v_1), \ldots, T(v_n)\}$ spans $\text{im}(T)$ .

Proof:

Any $w \in \text{im}(T)$ equals $T(v)$ for some $v = \sum \alpha_i v_i$ . Then:

w = T(v) = T\left(\sum \alpha_i v_i\right) = \sum \alpha_i T(v_i)

So $\{T(v_1), \ldots, T(v_n)\}$ spans $\text{im}(T)$ .

∎

Example 4.5: Finding Kernel and Image

Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be defined by $T(x, y, z) = (x + y, y + z)$ .

Kernel: $T(x, y, z) = (0, 0)$ gives:

x + y = 0, \quad y + z = 0

So $y = -x$ and $z = -y = x$ . Thus:

\ker(T) = \{(t, -t, t) : t \in \mathbb{R}\}

Image: Apply $T$ to the standard basis:

T(1,0,0) = (1,0), \quad T(0,1,0) = (1,1), \quad T(0,0,1) = (0,1)

So $\text{im}(T) = \text{span}\{(1,0), (1,1), (0,1)\} = \mathbb{R}^2$ (since these span $\mathbb{R}^2$ ).

Example 4.6: Computing via Matrix Representation

For a linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ given by $T(x) = Ax$ for matrix $A$ :

$\ker(T) = \{x : Ax = 0\}$ (null space of $A$ )
$\text{im}(T) = \{Ax : x \in \mathbb{R}^n\}$ (column space of $A$ )

Row reduce $A$ to find kernel (free variables) and image (pivot columns).

5. Composition of Linear Maps

Composing linear maps gives another linear map. Understanding composition is crucial for understanding how linear transformations interact and combine.

Theorem 5.1: Composition is Linear

If $T: U \to V$ and $S: V \to W$ are linear maps, then their composition $S \circ T: U \to W$ defined by $(S \circ T)(u) = S(T(u))$ is also linear.

Proof:

For $u_1, u_2 \in U$ and $\alpha \in F$ :

(S \circ T)(\alpha u_1 + u_2) = S(T(\alpha u_1 + u_2)) = S(\alpha T(u_1) + T(u_2)) = \alpha S(T(u_1)) + S(T(u_2)) = \alpha (S \circ T)(u_1) + (S \circ T)(u_2)

∎

Theorem 5.2: Kernel and Image of Composition

For linear maps $T: U \to V$ and $S: V \to W$ :

$\ker(T) \subseteq \ker(S \circ T)$
$\text{im}(S \circ T) \subseteq \text{im}(S)$
If $S$ is injective, then $\ker(S \circ T) = \ker(T)$
If $T$ is surjective, then $\text{im}(S \circ T) = \text{im}(S)$

Proof:

(1) If $u \in \ker(T)$ , then $T(u) = 0$ , so $(S \circ T)(u) = S(0) = 0$ . Thus $u \in \ker(S \circ T)$ .

(2) Any $w \in \text{im}(S \circ T)$ equals $S(T(u))$ for some $u$ , so $w \in \text{im}(S)$ .

(3) If $u \in \ker(S \circ T)$ , then $S(T(u)) = 0$ . Since $S$ is injective, $T(u) = 0$ , so $u \in \ker(T)$ .

(4) If $w \in \text{im}(S)$ , say $w = S(v)$ , and $T$ is surjective, then $v = T(u)$ for some $u$ , so $w = S(T(u)) \in \text{im}(S \circ T)$ .

∎

Example 5.1: Composition Example

Let $T: \mathbb{R}^2 \to \mathbb{R}^3$ be $T(x, y) = (x, y, 0)$ and $S: \mathbb{R}^3 \to \mathbb{R}^2$ be $S(x, y, z) = (x, y)$ .

Then $(S \circ T)(x, y) = S(x, y, 0) = (x, y)$ , so $S \circ T = I_{\mathbb{R}^2}$ (identity).

Note: $\ker(T) = \{0\}$ and $\ker(S \circ T) = \{0\}$ (consistent with (3)).

Theorem 5.3: Associativity of Composition

For linear maps $T: U \to V$ , $S: V \to W$ , $R: W \to X$ :

(R \circ S) \circ T = R \circ (S \circ T)

6. Linear Maps and Subspaces

Linear maps interact naturally with subspaces through preimages and restrictions. These constructions are fundamental for understanding the structure of linear transformations.

Definition 6.1: Preimage

For a linear map $T: V \to W$ and a subspace $U \subseteq W$ , the preimage (or inverse image) is:

T^{-1}(U) = \{v \in V : T(v) \in U\}

Theorem 6.1: Preimage of Subspace is Subspace

If $T: V \to W$ is linear and $U \subseteq W$ is a subspace, then $T^{-1}(U)$ is a subspace of $V$ .

Proof:

We verify the subspace criterion:

Contains zero: $T(0) = 0 \in U$ , so $0 \in T^{-1}(U)$
Closed under addition: If $v_1, v_2 \in T^{-1}(U)$ , then $T(v_1), T(v_2) \in U$ , so $T(v_1 + v_2) = T(v_1) + T(v_2) \in U$
Closed under scalar multiplication: If $v \in T^{-1}(U)$ and $\alpha \in F$ , then $T(\alpha v) = \alpha T(v) \in U$

∎

Definition 6.2: Restriction

For a linear map $T: V \to W$ and a subspace $U \subseteq V$ , the restriction of $T$ to $U$ is the map $T|_U: U \to W$ defined by $T|_U(u) = T(u)$ for all $u \in U$ .

Theorem 6.2: Properties of Restriction

For $T: V \to W$ and subspace $U \subseteq V$ :

$\ker(T|_U) = \ker(T) \cap U$
$\text{im}(T|_U) = T(U)$ (the image of the subspace $U$ under $T$ )

Example 6.1: Preimage Example

Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be $T(x, y, z) = (x, y)$ and $U = \{(0, t) : t \in \mathbb{R}\}$ (y-axis in $\mathbb{R}^2$ ).

Then $T^{-1}(U) = \{(0, y, z) : y, z \in \mathbb{R}\}$ (the yz-plane in $\mathbb{R}^3$ ).

Example 6.2: Restriction Example

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be $T(x, y, z) = (x, y, 0)$ (projection onto xy-plane).

Restrict to $U = \{(x, y, 0) : x, y \in \mathbb{R}\}$ (xy-plane). Then $T|_U = I_U$ (identity on $U$ ).

Theorem 6.3: Image of Subspace

If $T: V \to W$ is linear and $U \subseteq V$ is a subspace, then $T(U) = \{T(u) : u \in U\}$ is a subspace of $W$ .

Proof:

This is essentially the same as showing $\text{im}(T|_U)$ is a subspace, which follows from $T|_U$ being linear.

∎

Frequently Asked Questions

What's the difference between 'linear map', 'linear transformation', and 'linear operator'?

These terms are often used interchangeably, but with slight distinctions: 'Linear map' is the most general term for any linear function T: V → W. 'Linear transformation' is synonymous, though some authors prefer it when V = W. 'Linear operator' typically refers to linear maps from a space to itself (T: V → V), also called endomorphisms.

How do I check if a function is linear?

Method 1: Verify both properties separately: (a) T(u + v) = T(u) + T(v) for all u, v (additivity), and (b) T(αv) = αT(v) for all scalars α and vectors v (homogeneity). Method 2: Verify the combined condition: T(αu + βv) = αT(u) + βT(v) for all scalars α, β and vectors u, v. Quick check: If T(0) ≠ 0, then T is NOT linear.

Why must linear maps preserve the zero vector?

This follows from homogeneity: T(0) = T(0·v) = 0·T(v) = 0. This is a useful quick test: if T(0) ≠ 0, the map is definitely NOT linear.

How do kernel and image relate to matrices?

For T represented by matrix A: ker(T) = null space of A (solution space of Ax = 0), im(T) = column space of A. This connects abstract linear algebra to matrix computations and row reduction.

What's the geometric meaning of the kernel?

The kernel is what T 'collapses' to zero. For a projection onto a plane, the kernel is the line perpendicular to that plane. For differentiation, the kernel is constant functions. It measures the 'dimension loss' of the map.

Linear Maps & Kernel/Image Practice

10

Questions

0

Correct

0%

Accuracy

1

Is

T(x, y) = (x + y, x - y)

linear from

\mathbb{R}^2

to

\mathbb{R}^2

?

Easy

Not attempted

2

Is

T(x) = x + 1

linear from

\mathbb{R}

to

\mathbb{R}

?

Easy

Not attempted

3

What is

\ker(T)

where

T(x,y) = (x+y, x+y)

?

Medium

Not attempted

4

If

T

is linear and injective, what is

\dim(\ker T)

?

Easy

Not attempted

5

Is the image of a linear map always a subspace?

Easy

Not attempted

6

T: \mathbb{R}^3 \to \mathbb{R}^3

,

T(x,y,z) = (x,y,0)

. What is

\text{im}(T)

?

Medium

Not attempted

7

If

\text{im}(T) = W

, then

T

is:

Easy

Not attempted

8

What is

\ker(I_V)

where

I_V

is the identity on

V

?

Easy

Not attempted

9

If

\ker(T) \neq \{0\}

, then

T

is:

Medium

Not attempted

10

If

\dim V = 5

and

\dim(\ker T) = 2

, what is

\dim(\text{im } T)

?

Medium

Not attempted