LA-3.1

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Linear Map Definition

A linear map is a function between vector spaces that preserves the vector space structure—it respects both addition and scalar multiplication. This simple property underlies virtually all of linear algebra.

3-4 hours Core Level 10 Objectives

Learning Objectives

State the definition of a linear map between vector spaces
Verify whether a given function is linear using the definition
Recognize standard examples: differentiation, integration, projection, rotation
Understand the decomposition into additivity and homogeneity
Prove basic properties: T(0) = 0, T(-v) = -T(v)
Define and work with the space of linear maps ℒ(V, W)
Compute dimensions of spaces of linear maps
Understand and verify composition of linear maps
Define and recognize linear functionals
Distinguish linear from non-linear maps with counterexamples

Prerequisites

Vector space definition (LA-2.1)
Basis and dimension (LA-2.4)
Basic properties of vector spaces
Function composition
Field axioms

Historical Context

The concept of a linear map evolved alongside linear algebra itself. Arthur Cayley (1821–1895) introduced matrix algebra in 1858, implicitly using linear transformations. Giuseppe Peano (1858–1932) gave the first abstract definition of vector spaces and linear operations in 1888.

The modern "linear maps before matrices" approach emphasizes that linear maps are the fundamental objects and matrices are merely their representations with respect to chosen bases.

Linear maps appear throughout mathematics: differentiation and integration in calculus, expected value in probability, Fourier transforms in analysis. The linearity property underlies much of modern physics, from quantum mechanics to signal processing.

1. Definition of Linear Maps

The concept of a linear map is central to all of linear algebra. A linear map is a function between vector spaces that respects the vector space structure—it preserves addition and scalar multiplication. This property makes linear maps remarkably well-behaved and leads to a rich theory.

Definition 3.1: 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 3.1: 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 $v \in V$

These two conditions together are equivalent to the single condition in Definition 3.1. Having both conditions is sometimes more convenient for verification.

Theorem 3.1: 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 $v, w \in V$
$T\left(\sum_{i=1}^{n} \alpha_i v_i\right) = \sum_{i=1}^{n} \alpha_i T(v_i)$ for any finite linear combination

Proof:

(1) Using homogeneity with $\alpha = 0$ :

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

(2) Using homogeneity with $\alpha = -1$ :

T(-v) = T((-1) \cdot v) = (-1) \cdot T(v) = -T(v)

(3) Combining additivity and (2):

T(v - w) = T(v + (-w)) = T(v) + T(-w) = T(v) - T(w)

(4) Follows by induction on $n$ , using additivity and homogeneity repeatedly.

∎

Example 3.1: Standard Examples of Linear Maps

Here are fundamental examples that appear throughout mathematics:

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 $v \in V$
Scalar Multiplication: $\lambda I: V \to V$ defined by $(\lambda I)(v) = \lambda v$
Differentiation: $D: P_n(\mathbb{R}) \to P_{n-1}(\mathbb{R})$ defined by $D(p) = p'$
Integration: $\int_0^x: P_n(\mathbb{R}) \to P_{n+1}(\mathbb{R})$ defined by $p \mapsto \int_0^x p(t)\,dt$

Example 3.2: Geometric Linear Maps on ℝ²

Geometric transformations on $\mathbb{R}^2$ provide intuitive examples:

Rotation by angle θ: $R_\theta(x, y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)$
Projection onto x-axis: $\pi(x, y) = (x, 0)$
Reflection about x-axis: $\rho(x, y) = (x, -y)$
Scaling: $S_c(x, y) = (cx, cy)$ for scalar $c$
Shear: $H_k(x, y) = (x + ky, y)$

Example 3.3: Verifying Linearity

Claim: The map $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $T(x_1, x_2) = (x_1 - x_2, x_1, x_1 + x_2)$ is linear.

Proof: Let $(x_1, x_2), (y_1, y_2) \in \mathbb{R}^2$ and $k_1, k_2 \in \mathbb{R}$ .

T(k_1(x_1, x_2) + k_2(y_1, y_2)) = T(k_1 x_1 + k_2 y_1, k_1 x_2 + k_2 y_2)

= ((k_1 x_1 + k_2 y_1) - (k_1 x_2 + k_2 y_2), k_1 x_1 + k_2 y_1, (k_1 x_1 + k_2 y_1) + (k_1 x_2 + k_2 y_2))

= k_1(x_1 - x_2, x_1, x_1 + x_2) + k_2(y_1 - y_2, y_1, y_1 + y_2)

= k_1 T(x_1, x_2) + k_2 T(y_1, y_2)

Example 3.4: 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: $T(x_1, x_2) = (x_1 x_2, x_1 + x_2)$ . Check: $T((1,0) + (0,1)) = T(1,1) = (1, 2)$ but $T(1,0) + T(0,1) = (0,1) + (0,1) = (0,2)$ .
Determinant: $\det: M_n(\mathbb{R}) \to \mathbb{R}$ . Since $\det(2I) = 2^n \neq 2\det(I)$ for $n > 1$ .
Norm: $\|\cdot\|: V \to \mathbb{R}$ . Since $\|-v\| = \|v\| \neq -\|v\|$ .

2. The Space of Linear Maps ℒ(V, W)

One of the beautiful features of linear algebra is that linear maps themselves form a vector space. This allows us to add linear maps and multiply them by scalars, creating new linear maps.

Definition 3.2: Space of Linear Maps

Let $V$ and $W$ be vector spaces over the same field $F$ . We denote by $\mathcal{L}(V, W)$ the set of all linear maps from $V$ to $W$ .

When $V = W$ , we write $\mathcal{L}(V)$ for $\mathcal{L}(V, V)$ , and call its elements linear operators on $V$ .

Definition 3.3: Operations on Linear Maps

For $S, T \in \mathcal{L}(V, W)$ and $\lambda \in F$ , we define:

Sum: $(S + T)(v) = S(v) + T(v)$ for all $v \in V$
Scalar multiple: $(\lambda T)(v) = \lambda \cdot T(v)$ for all $v \in V$

Theorem 3.2: ℒ(V, W) is a Vector Space

With addition and scalar multiplication defined as above, $\mathcal{L}(V, W)$ is a vector space over $F$ . The zero vector is the zero map $0$ , and the additive inverse of $T$ is $-T$ .

Proof:

We verify that $S + T$ and $\lambda T$ are linear maps:

Sum is linear:

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

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

Scalar multiple is linear:

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

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

The vector space axioms follow from the corresponding properties in $W$ .

∎

Theorem 3.3: Dimension of ℒ(V, W)

If $\dim V = m$ and $\dim W = n$ are finite, then:

\dim \mathcal{L}(V, W) = m \cdot n

Proof:

A linear map $T: V \to W$ is completely determined by its values on a basis of $V$ . If $\{v_1, \ldots, v_m\}$ is a basis of $V$ , then $T(v_j)$ can be any vector in $W$ . Each $T(v_j)$ requires $n$ coordinates to specify, and there are $m$ basis vectors, giving $mn$ degrees of freedom.

∎

Example 3.5: Dimension Calculation

$\dim \mathcal{L}(\mathbb{R}^3, \mathbb{R}^2) = 3 \times 2 = 6$
$\dim \mathcal{L}(\mathbb{R}^n, \mathbb{R}) = n \times 1 = n$
$\dim \mathcal{L}(\mathbb{R}^n) = n^2$

3. Composition of Linear Maps

When we compose two linear maps, the result is again linear. This composition operation corresponds to matrix multiplication when we represent maps as matrices.

Definition 3.4: Composition

Let $T \in \mathcal{L}(V, W)$ and $S \in \mathcal{L}(W, U)$ . The composition $S \circ T \in \mathcal{L}(V, U)$ is defined by:

(S \circ T)(v) = S(T(v)) \quad \text{for all } v \in V

Theorem 3.4: Composition is Linear

If $T \in \mathcal{L}(V, W)$ and $S \in \mathcal{L}(W, U)$ , then $S \circ T \in \mathcal{L}(V, U)$ .

Proof:

For all $u, v \in V$ and $\alpha, \beta \in F$ :

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

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

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

∎

Theorem 3.5: Properties of Composition

Composition of linear maps satisfies:

Associativity: $(R \circ S) \circ T = R \circ (S \circ T)$
Identity: $I_W \circ T = T = T \circ I_V$
Distributivity: $S \circ (T_1 + T_2) = S \circ T_1 + S \circ T_2$
Distributivity: $(S_1 + S_2) \circ T = S_1 \circ T + S_2 \circ T$
Scalar compatibility: $\lambda(S \circ T) = (\lambda S) \circ T = S \circ (\lambda T)$

Remark 3.2: Non-commutativity

In general, $S \circ T \neq T \circ S$ ! This is one of the key differences from ordinary number multiplication. Even when both compositions are defined (e.g., for operators on $V$ ), they may produce different results.

Example 3.6: Non-commuting Compositions

Let $D: P_3(\mathbb{R}) \to P_2(\mathbb{R})$ be differentiation and $M: P_2(\mathbb{R}) \to P_3(\mathbb{R})$ be multiplication by $x$ .

For $p(x) = x^2$ :

$(D \circ M)(x^2) = D(x^3) = 3x^2$
$(M \circ D)(x^2) = M(2x) = 2x^2$

So $D \circ M \neq M \circ D$ .

4. Linear Maps are Determined by Basis Values

One of the most powerful results about linear maps is that they are completely determined by their action on a basis. This is the foundation for representing linear maps as matrices.

Theorem 3.6: Determination by Basis

Let $\{v_1, \ldots, v_n\}$ be a basis of $V$ . If $T, S \in \mathcal{L}(V, W)$ satisfy $T(v_i) = S(v_i)$ for all $i = 1, \ldots, n$ , then $T = S$ .

Proof:

For any $v \in V$ , we can write $v = \sum_{i=1}^{n} \alpha_i v_i$ . Then:

T(v) = T\left(\sum_{i=1}^{n} \alpha_i v_i\right) = \sum_{i=1}^{n} \alpha_i T(v_i) = \sum_{i=1}^{n} \alpha_i S(v_i) = S(v)

Since this holds for all $v \in V$ , we have $T = S$ .

∎

Theorem 3.7: Existence and Uniqueness

Let $\{v_1, \ldots, v_n\}$ be a basis of $V$ , and let $w_1, \ldots, w_n$ be any vectors in $W$ . Then there exists a unique linear map $T \in \mathcal{L}(V, W)$ such that $T(v_i) = w_i$ for all $i = 1, \ldots, n$ .

Proof:

Existence: Define $T$ by

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

This is well-defined since every $v \in V$ has a unique representation as a linear combination of basis vectors. One can verify directly that $T$ is linear.

Uniqueness: Follows from Theorem 3.6.

∎

Example 3.7: Constructing a Linear Map

Problem: Does there exist a linear map $T: \mathbb{R}^3 \to \mathbb{R}^2$ with $T(1, -1, 1) = (1, 0)$ and $T(1, 1, 1) = (0, 1)$ ?

Solution: Yes! Extend $\{(1,-1,1), (1,1,1)\}$ to a basis of $\mathbb{R}^3$ by adding $(1, 0, 0)$ . Set $T(1, 0, 0) = (0, 0)$ (arbitrary choice). By Theorem 3.7, a unique linear map exists.

Remark 3.3: When Does a Linear Map NOT Exist?

A proposed linear map fails to exist when:

The given conditions would require $T(0) \neq 0$
The given conditions would map dependent vectors to independent vectors
The given conditions are inconsistent (e.g., same input, different outputs)

5. Linear Functionals

Linear functionals are linear maps to the scalar field. They form the dual space and are fundamental in many areas of mathematics.

Definition 3.5: Linear Functional

A linear functional (or linear form) on $V$ is a linear map $\varphi: V \to F$ , where $F$ is the scalar field of $V$ .

Example 3.8: Examples of Linear Functionals

Evaluation: For polynomials, $\varphi_a(p) = p(a)$ evaluates at $a$
Integration: $\varphi(f) = \int_a^b f(x)\,dx$ on $C[a,b]$
Trace: $\text{tr}(A) = \sum_{i=1}^{n} a_{ii}$ on $M_n(F)$
Coordinate functional: $\pi_i(x_1, \ldots, x_n) = x_i$ on $F^n$

Theorem 3.8: Linear Functionals on Finite-Dimensional Spaces

If $\dim V = n$ , then $\dim V^* = \dim \mathcal{L}(V, F) = n$ .

6. Worked Examples

Example 1: Verify Linearity

Problem: Is $T: P_2(\mathbb{R}) \to P_1(\mathbb{R})$ defined by $T(p) = p(x+1) - p(x)$ linear?

Solution: For $p_1, p_2 \in P_2$ and $k_1, k_2 \in \mathbb{R}$ :

T(k_1 p_1 + k_2 p_2) = (k_1 p_1 + k_2 p_2)(x+1) - (k_1 p_1 + k_2 p_2)(x)

= k_1(p_1(x+1) - p_1(x)) + k_2(p_2(x+1) - p_2(x)) = k_1 T(p_1) + k_2 T(p_2)

✓ T is linear.

Example 2: Find dim ℒ(V, W)

Problem: Find $\dim \mathcal{L}(P_3(\mathbb{R}), M_{2 \times 2}(\mathbb{R}))$ .

Solution:

$\dim P_3(\mathbb{R}) = 4$ (basis: $\{1, x, x^2, x^3\}$ )
$\dim M_{2 \times 2}(\mathbb{R}) = 4$

Therefore, $\dim \mathcal{L}(P_3, M_{2 \times 2}) = 4 \times 4 = 16$ .

Example 3: Disprove Linearity

Problem: Show $T(x_1, x_2) = (x_1^2, x_2)$ is not linear.

Solution: Check homogeneity:

T(2 \cdot (1, 0)) = T(2, 0) = (4, 0)

2 \cdot T(1, 0) = 2 \cdot (1, 0) = (2, 0)

Since $(4, 0) \neq (2, 0)$ , T is NOT linear.

Example 4: Composition

Problem: Let $S(x, y) = (x + y, x)$ and $T(x, y) = (2x, y - x)$ . Find $S \circ T$ and $T \circ S$ .

Solution:

(S \circ T)(x, y) = S(T(x, y)) = S(2x, y-x) = (2x + y - x, 2x) = (x + y, 2x)

(T \circ S)(x, y) = T(S(x, y)) = T(x+y, x) = (2(x+y), x - (x+y)) = (2x+2y, -y)

Note: $S \circ T \neq T \circ S$ .

Example 5: Existence Question

Problem: Does there exist a linear map $T: \mathbb{R}^2 \to \mathbb{R}^3$ with

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

Solution: No! If $T$ were linear:

T(1, 1) = T((1,0) + (0,1)) = T(1,0) + T(0,1) = (1,0,0) + (0,1,0) = (1, 1, 0)

But the condition says $T(1, 1) = (0, 0, 1) \neq (1, 1, 0)$ . Contradiction!

Example 6: Linear Functional

Problem: Show that $\varphi: M_2(\mathbb{R}) \to \mathbb{R}$ given by $\varphi(A) = \text{tr}(A)$ is a linear functional.

Solution: For matrices $A, B$ and scalar $c$ :

\text{tr}(A + B) = \sum_i (a_{ii} + b_{ii}) = \sum_i a_{ii} + \sum_i b_{ii} = \text{tr}(A) + \text{tr}(B)

\text{tr}(cA) = \sum_i c \cdot a_{ii} = c \sum_i a_{ii} = c \cdot \text{tr}(A)

✓ Trace is a linear functional.

7. Common Mistakes

Mistake 1: Forgetting T(0) = 0

If $T(0) \neq 0$ , then $T$ is definitely NOT linear. This is a quick first check when testing linearity.

Mistake 2: Assuming Composition Commutes

$S \circ T \neq T \circ S$ in general! Even for operators on the same space, composition order matters.

Mistake 3: Confusing Linear with Affine

$T(x) = ax + b$ is NOT linear if $b \neq 0$ . This is an affine map. Linear maps through the origin: $T(x) = ax$ .

Mistake 4: Dependent to Independent

Linear maps preserve dependence but not independence. If vectors are dependent, their images must be dependent. But independent vectors can map to dependent ones.

Mistake 5: Checking Only One Property

Both additivity AND homogeneity must hold. Some maps satisfy one but not the other (especially over certain fields).

8. Key Takeaways

Definition

$T(\alpha u + \beta v) = \alpha T(u) + \beta T(v)$ for all vectors and scalars. Equivalently: additivity + homogeneity.

Basic Properties

$T(0) = 0$ , $T(-v) = -T(v)$ , and linear maps preserve linear combinations and dependence.

Space of Maps

$\mathcal{L}(V, W)$ is a vector space with $\dim = (\dim V)(\dim W)$ . Composition is associative but not commutative.

Basis Determination

A linear map is uniquely determined by its values on a basis. Given basis values, there exists a unique linear extension.

9. Connection to the Rest of Linear Algebra

Linear maps are the central objects of study in linear algebra. They connect to every major topic in the subject and provide the framework for understanding matrices, determinants, eigenvalues, and more.

Matrices as Representations

Every linear map $T: V \to W$ between finite-dimensional spaces can be represented by a matrix once we choose bases for $V$ and $W$ . The matrix encodes the images of basis vectors as its columns.

[T]_{\mathcal{B}, \mathcal{C}} = \text{matrix of } T \text{ with respect to bases } \mathcal{B} \text{ and } \mathcal{C}

Kernel and Image

Two fundamental subspaces associated with a linear map $T: V \to W$ :

Kernel (null space): $\ker(T) = \{v \in V : T(v) = 0\}$
Image (range): $\text{im}(T) = \{T(v) : v \in V\}$

The Rank-Nullity Theorem relates these: $\dim(\ker T) + \dim(\text{im } T) = \dim V$ .

Invertibility and Isomorphism

A linear map is an isomorphism if it is bijective. For linear maps between spaces of the same dimension, the following are equivalent:

$T$ is injective ( $\ker T = \{0\}$ )
$T$ is surjective ( $\text{im } T = W$ )
$T$ is bijective (invertible)

Eigenvalues and Eigenvectors

For a linear operator $T: V \to V$ , an eigenvector is a nonzero vector $v$ such that $T(v) = \lambda v$ for some scalar $\lambda$ (the eigenvalue).

Eigenvalues reveal the intrinsic behavior of linear operators and are essential for diagonalization, solving differential equations, and many applications.

Dual Spaces

The dual space $V^* = \mathcal{L}(V, F)$ consists of all linear functionals on $V$ . Every linear map $T: V \to W$ induces a dual map $T^*: W^* \to V^*$ .

The dual perspective provides powerful tools for understanding linear algebra through row operations, transpose matrices, and annihilators.

10. Advanced Examples

Example: Rotation is Linear

Problem: Prove that the rotation $R_\theta: \mathbb{R}^2 \to \mathbb{R}^2$ by angle $\theta$ is linear.

Solution: The rotation is given by:

R_\theta(x, y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)

For any vectors $(x_1, y_1), (x_2, y_2)$ and scalars $\alpha, \beta$ :

R_\theta(\alpha(x_1, y_1) + \beta(x_2, y_2)) = R_\theta(\alpha x_1 + \beta x_2, \alpha y_1 + \beta y_2)

= ((\alpha x_1 + \beta x_2)\cos\theta - (\alpha y_1 + \beta y_2)\sin\theta, ...)

= \alpha(x_1\cos\theta - y_1\sin\theta, x_1\sin\theta + y_1\cos\theta) + \beta(...)

= \alpha R_\theta(x_1, y_1) + \beta R_\theta(x_2, y_2)

Example: Differentiation Operator

Problem: Show that $D: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ defined by $D(f) = f'$ is linear.

Solution: For differentiable functions $f, g$ and scalars $\alpha, \beta$ :

D(\alpha f + \beta g) = (\alpha f + \beta g)' = \alpha f' + \beta g' = \alpha D(f) + \beta D(g)

This follows from the linearity of differentiation in calculus.

Example: Expected Value is Linear

Problem: Show that expected value $\mathbb{E}: V \to \mathbb{R}$ on a space of random variables is linear.

Solution: The fundamental property of expected value is:

\mathbb{E}[\alpha X + \beta Y] = \alpha \mathbb{E}[X] + \beta \mathbb{E}[Y]

This is exactly the linearity condition! Expected value is a linear functional.

Example: Transpose as a Linear Map

Problem: Show that $T: M_{m \times n}(\mathbb{R}) \to M_{n \times m}(\mathbb{R})$ defined by $T(A) = A^T$ is linear.

Solution: For matrices $A, B$ and scalars $\alpha, \beta$ :

T(\alpha A + \beta B) = (\alpha A + \beta B)^T = \alpha A^T + \beta B^T = \alpha T(A) + \beta T(B)

This uses the properties $(A + B)^T = A^T + B^T$ and $(\alpha A)^T = \alpha A^T$ .

Example: Shift Operator

Problem: On the space of sequences $\mathbb{R}^\infty$ , define the right shift $R(x_1, x_2, x_3, \ldots) = (0, x_1, x_2, \ldots)$ . Is this linear?

Solution: Yes! For sequences $(x_n), (y_n)$ and scalars $\alpha, \beta$ :

R(\alpha(x_n) + \beta(y_n)) = R(\alpha x_1 + \beta y_1, \alpha x_2 + \beta y_2, \ldots)

= (0, \alpha x_1 + \beta y_1, \alpha x_2 + \beta y_2, \ldots)

= \alpha(0, x_1, x_2, \ldots) + \beta(0, y_1, y_2, \ldots) = \alpha R(x_n) + \beta R(y_n)

11. Theoretical Insights

Theorem 3.9: Preservation of Linear Dependence

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

Proof:

Since $\{v_1, \ldots, v_k\}$ is dependent, there exist scalars $\alpha_1, \ldots, \alpha_k$ , not all zero, such that $\sum \alpha_i v_i = 0$ . Applying $T$ :

\sum_{i=1}^k \alpha_i T(v_i) = T\left(\sum_{i=1}^k \alpha_i v_i\right) = T(0) = 0

The same (non-all-zero) scalars show that $\{T(v_1), \ldots, T(v_k)\}$ is linearly dependent.

∎

Remark 3.4: Independence May Not Be Preserved

The converse is false! A linear map can take linearly independent vectors to linearly dependent vectors. For example, the projection $\pi(x, y) = (x, 0)$ maps the independent set $\{(1, 0), (0, 1)\}$ to $\{(1, 0), (0, 0)\}$ , which is dependent.

Theorem 3.10: Image of a Basis Spans the Image

If $\{v_1, \ldots, v_n\}$ is a basis of $V$ and $T: V \to W$ is linear, then:

\text{im}(T) = \text{span}\{T(v_1), \ldots, T(v_n)\}

Proof:

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

w = T(v) = T\left(\sum \alpha_i v_i\right) = \sum \alpha_i T(v_i) \in \text{span}\{T(v_1), \ldots, T(v_n)\}

Conversely, any element of the span is in the image since each $T(v_i) \in \text{im}(T)$ .

∎

Corollary 3.1: Rank Bound

For any linear map $T: V \to W$ with $\dim V = n$ :

\text{rank}(T) = \dim(\text{im } T) \leq \min\{\dim V, \dim W\}

Theorem 3.11: Linear Maps and Subspaces

Let $T: V \to W$ be linear and $U$ be a subspace of $V$ . Then:

$T(U)$ is a subspace of $W$
If $U' \leq W$ , then $T^{-1}(U') = \{v \in V : T(v) \in U'\}$ is a subspace of $V$

12. Additional Practice Problems

Problem 1

Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be defined by $T(x, y, z) = (x + y, 2z - x)$ . Verify that $T$ is linear and find $T(1, 2, 3)$ .

Problem 2

Prove or disprove: The map $T: M_{2 \times 2}(\mathbb{R}) \to M_{2 \times 2}(\mathbb{R})$ defined by $T(A) = A^2$ is linear.

Problem 3

Find $\dim \mathcal{L}(P_4(\mathbb{R}), \mathbb{R}^3)$ .

Problem 4

Let $T, S: V \to V$ be linear operators. Prove that $T + S$ and $T \circ S$ are also linear operators on $V$ .

Problem 5

Does there exist a linear map $T: \mathbb{R}^2 \to \mathbb{R}^2$ such that $T(1, 1) = (2, 3)$ and $T(2, 2) = (1, 1)$ ? Justify your answer.

Problem 6

Let $V = \{p \in P_3(\mathbb{R}) : p(0) = 0\}$ . Show that $V$ is a vector space and that the evaluation map $\varphi_1: V \to \mathbb{R}$ defined by $\varphi_1(p) = p(1)$ is a linear functional.

13. Quick Reference Summary

Concept	Definition/Formula
Linear Map	$T(\alpha u + \beta v) = \alpha T(u) + \beta T(v)$
Zero Property	$T(0) = 0$ always
Space of Maps	$\mathcal{L}(V, W)$ , dim = $(\dim V)(\dim W)$
Sum of Maps	$(S + T)(v) = S(v) + T(v)$
Scalar Multiple	$(\lambda T)(v) = \lambda T(v)$
Composition	$(S \circ T)(v) = S(T(v))$ , generally $S \circ T \neq T \circ S$
Linear Functional	$\varphi: V \to F$ , where $F$ is the scalar field
Basis Determination	$T$ is uniquely determined by values on any basis

What's Next?

Now that you understand what linear maps are, the next steps in our journey explore:

Kernel and Image: The fundamental subspaces associated with every linear map
Rank-Nullity Theorem: The dimension formula connecting kernel and image
Isomorphisms: When two vector spaces are "structurally identical"
Matrix Representation: How to represent linear maps as matrices
Dual Spaces: The vector space of linear functionals

These concepts build directly on the definition of linear maps and reveal the deep structure underlying linear algebra.

Linear Map Definition Practice

Questions

Correct

Accuracy

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

linear from

\mathbb{R}^2

\mathbb{R}^2

Easy

Not attempted

T(x) = x + 1

linear from

\mathbb{R}

\mathbb{R}

Easy

Not attempted

What is

\dim(\mathcal{L}(\mathbb{R}^2, \mathbb{R}^3))

Medium

Not attempted

T: V \to W

and

S: W \to U

are linear, is

S \circ T

linear?

Medium

Not attempted

Is differentiation

D: P_n \to P_{n-1}

defined by

D(p) = p'

linear?

Easy

Not attempted

The zero map

0: V \to W

defined by

0(v) = 0_W

for all

v

is:

Easy

Not attempted

T(x_1, x_2) = (x_1 x_2, x_1 + x_2)

linear from

\mathbb{R}^2

\mathbb{R}^2

Medium

Not attempted

T(f) = f(0)

linear from

C[0,1]

\mathbb{R}

Medium

Not attempted

The rotation

R_\theta: \mathbb{R}^2 \to \mathbb{R}^2

by angle

\theta

is:

Medium

Not attempted

T: V \to W

is linear and

\{v_1, \ldots, v_n\}

is linearly dependent, then

\{T(v_1), \ldots, T(v_n)\}

is:

Hard

Not attempted

Is the map

T: M_{2\times 2}(\mathbb{R}) \to \mathbb{R}

given by

T(A) = \det(A)

linear?

Hard

Not attempted

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

is linear and

T(e_1) = T(e_2) = T(e_3) = v

for some

v

, what is

T(1, 2, 3)

Hard

Not attempted

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. 'Linear functional' specifically means a linear map to the scalar field (T: V → F).

Why is linearity such an important property?

Linearity means the map respects the vector space structure—it preserves addition and scalar multiplication. This has profound consequences: (1) A linear map is completely determined by its action on a basis, (2) Linear maps can be represented by matrices, (3) Composition of linear maps is linear, (4) The set of linear maps forms a vector space itself, (5) Many important operations (differentiation, integration, expected value) are linear.

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.

Can linear maps between infinite-dimensional spaces be represented by matrices?

Not in the usual finite sense. For infinite-dimensional spaces, you would need 'infinite matrices,' but these require careful handling of convergence. In functional analysis, continuous linear maps between Banach or Hilbert spaces are studied using operator theory, which generalizes the finite-dimensional theory.

What are some common non-examples of linear maps?

Common non-linear maps include: (1) Translation: f(x) = x + c for c ≠ 0 (fails T(0) = 0), (2) Squaring: f(x) = x² (not homogeneous), (3) Absolute value: f(x) = |x| (not homogeneous), (4) Determinant: det(cA) = c^n det(A) ≠ c·det(A) for n > 1, (5) Any function with a constant term.

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.

What is a linear functional?

A linear functional (or linear form) is a linear map from a vector space V to its scalar field F: f: V → F. Examples include evaluation at a point f(p) = p(a), integration f(g) = ∫g(x)dx, and trace tr(A). Linear functionals form the dual space V*.

How does the dimension formula dim(ℒ(V,W)) = mn arise?

A linear map T: V → W is completely determined by its values on a basis of V. If dim(V) = m and dim(W) = n, then T(vⱼ) can be any vector in W for each basis vector vⱼ. We need n scalars for each of m basis vectors, giving mn degrees of freedom total.

Is the composition of linear maps always defined?

No! For S ∘ T to be defined, the codomain of T must equal the domain of S. If T: V → W and S: W → U, then S ∘ T: V → U is defined and linear. But if the spaces don't match, composition is undefined.

Can a linear map take linearly independent vectors to linearly dependent vectors?

Yes! For example, T(x, y) = (x + y, x + y) takes the independent set {(1, 0), (0, 1)} to the dependent set {(1, 1), (1, 1)}. Linear maps preserve dependence but not necessarily independence.