Linear Transformations: Definition & Properties

🎯 Learning Objectives

• State the two defining properties of a linear transformation
• Determine whether a given map is linear
• Identify the standard matrix of a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$
• Describe geometric transformations (rotation, reflection, scaling, shear) as linear maps
• Prove basic properties of linear transformations

📋 Prerequisites

• Vector Spaces (Week 3) — the domain and codomain are vector spaces
• Matrix Multiplication (Week 1) — the computational representation of linear maps
• Linear transformations are the functions of linear algebra — they move vectors between spaces while preserving structure

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1. Intuition: Structure-Preserving Maps

1.1 What Makes a Map "Linear"?

• $f(x+y) = 2(x+y) = 2x + 2y = f(x) + f(y)$
• $f(cx) = 2(cx) = c(2x) = cf(x)$ These two properties — additivity and homogeneity — capture what it means to be linear. A linear transformation is a function between vector spaces that respects addition and scalar multiplication. It's the natural notion of "function" in the world of vector spaces. (Diagram)

1.2 Why This Matters

• Understand them completely by examining a few vectors (basis vectors)
• Represent them as matrices (once we choose bases)
• Classify them by their kernel and image

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2. Definition

Definition (Linear Transformation). A function $T: V \to W$ between two vector spaces $V$ and $W$ (over $\mathbb{R}$) is a linear transformation if:

1. Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ for all $\mathbf{u}, \mathbf{v} \in V$
2. Homogeneity: $T(c\mathbf{v}) = cT(\mathbf{v})$ for all $c \in \mathbb{R}$, $\mathbf{v} \in V$

Equivalently: $T(c\mathbf{u} + d\mathbf{v}) = cT(\mathbf{u}) + dT(\mathbf{v})$ for all scalars $c, d$.

2.1 Terminology

2.2 Immediate Consequences

• $T(\mathbf{0}_V) = \mathbf{0}_W$ (prove: $T(0\cdot\mathbf{v}) = 0\cdot T(\mathbf{v}) = \mathbf{0}$)
• $T(-\mathbf{v}) = -T(\mathbf{v})$
• $T(\sum_{i=1}^k c_i \mathbf{v}_i) = \sum_{i=1}^k c_i T(\mathbf{v}_i)$ (linearity extends to any finite combination)

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3. Examples

3.1 The Zero Transformation

3.2 The Identity Transformation

3.3 Scaling (Dilation)

• $T(\mathbf{u}+\mathbf{v}) = c(\mathbf{u}+\mathbf{v}) = c\mathbf{u} + c\mathbf{v} = T(\mathbf{u}) + T(\mathbf{v})$
• $T(d\mathbf{v}) = c(d\mathbf{v}) = d(c\mathbf{v}) = dT(\mathbf{v})$

3.4 Rotation in $\mathbb{R}^2$

$$T\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix} x\cos\theta - y\sin\theta \\ x\sin\theta + y\cos\theta \end{pmatrix}$$ $$\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$

Example 1: Rotation by $90^\circ$

$\theta = 90^\circ = \pi/2$: $\cos\theta = 0$, $\sin\theta = 1$.

$$> T(x,y) = (-y, x) >$$

Check linearity: $T((x_1,y_1)+(x_2,y_2)) = (-(y_1+y_2), x_1+x_2) = (-y_1, x_1) + (-y_2, x_2) = T(x_1,y_1) + T(x_2,y_2)$ ✓

3.5 Reflection Across $x$-axis

$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

3.6 Projection onto $x$-axis

$$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$$

3.7 Differentiation

Example 2: Differentiation as a linear transformation

$T(2 + 3x - x^2) = 3 - 2x$. $T(1 + x^2) = 2x$. $T((2+3x-x^2) + (1+x^2)) = T(3 + 3x) = 3 = (3-2x) + 2x = T(2+3x-x^2) + T(1+x^2)$ ✓

3.8 Integration

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4. Non-Examples

4.1 $T(x) = x^2$ (from $\mathbb{R}$ to $\mathbb{R}$)

4.2 $T(x) = x + 1$ (affine, not linear)

4.3 $T(x,y) = (|x|, y)$

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5. Matrix Representation

5.1 Standard Matrix

$$T(\mathbf{x}) = A\mathbf{x}$$ $$A = [T(\mathbf{e}_1) \; T(\mathbf{e}_2) \; \dots \; T(\mathbf{e}_n)]$$

Example 3: Finding the standard matrix

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

$T(\mathbf{e}_1) = T(1,0,0) = (2, 1)$ $T(\mathbf{e}_2) = T(0,1,0) = (-1, 3)$ $T(\mathbf{e}_3) = T(0,0,1) = (0, 1)$

$$> A = \begin{bmatrix} 2 & -1 & 0 \\ 1 & 3 & 1 \end{bmatrix} >$$

Check:

$$> A\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}2x - y \\ x + 3y + z\end{bmatrix} >$$

✓

5.2 Composition as Matrix Multiplication

$$(S \circ T)(\mathbf{x}) = S(T(\mathbf{x})) = B(A\mathbf{x}) = (BA)\mathbf{x}$$

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6. Types of Linear Transformations

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7. Edge Cases & Gotchas

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8. Common Pitfalls

❌ Pitfall 1: Checking only additivity

❌ Pitfall 2: Thinking all functions are linear

❌ Pitfall 3: Forgetting to check domain/codomain are vector spaces

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9. Formula Summary Table

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10. 📝 Practice Questions

Q1: Is it linear?

Is $T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x,y) = (x+1, y+2)$ linear?

Solution: $T(0,0) = (1,2) \neq (0,0)$. Not linear. Q2: Is it linear?

Is $T: \mathbb{R}^3 \to \mathbb{R}$, $T(x,y,z) = 2x - 3y + z$ linear?

Solution: Check additivity: $T((x_1,y_1,z_1)+(x_2,y_2,z_2)) = 2(x_1+x_2) - 3(y_1+y_2) + (z_1+z_2)$ $= (2x_1-3y_1+z_1) + (2x_2-3y_2+z_2) = T(x_1,y_1,z_1) + T(x_2,y_2,z_2)$ ✓

Check homogeneity: $T(cx,cy,cz) = 2cx - 3cy + cz = c(2x-3y+z) = cT(x,y,z)$ ✓

Yes, it is linear. Q3: Find the standard matrix

Find the matrix of $T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x,y) = (2x-3y, x+4y)$.

Solution: $T(\mathbf{e}_1) = T(1,0) = (2, 1)$ $T(\mathbf{e}_2) = T(0,1) = (-3, 4)$

$$> A = \begin{bmatrix} 2 & -3 \\ 1 & 4 \end{bmatrix} >$$

. Q4: Rotation matrix

Find the matrix for rotation by $60^\circ$ ($\pi/3$).

Solution: $\cos 60^\circ = \frac{1}{2}$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$.

$$> A = \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix} >$$

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Check: rotates $(1,0)$ to $(\frac{1}{2}, \frac{\sqrt{3}}{2})$, which is $(1,0)$ rotated $60^\circ$ ✓ Q5: Composition

$T: \mathbb{R}^2 \to \mathbb{R}^2$ rotates by $90^\circ$, $S: \mathbb{R}^2 \to \mathbb{R}^2$ reflects across $x$-axis. Find the matrix of $S \circ T$.

Solution: $T$ matrix:

$$> A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} >$$

$S$ matrix:

$$> B = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} >$$

$S \circ T$ matrix:

$$> BA = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} >$$

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Check: $(S \circ T)(1,0) = S(0,1) = (0,-1)$.

$$> BA(1,0)^T = \begin{bmatrix} 0&-1\\-1&0 \end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}0\\-1\end{bmatrix} >$$

✓ Q6: Proving linearity

Show that $T: P_1 \to P_2$, $T(p)(x) = \int_0^x p(t) dt$ is linear.

Solution: Let $p, q \in P_1$, $c \in \mathbb{R}$. $T(p+q)(x) = \int_0^x (p+q)(t) dt = \int_0^x (p(t)+q(t)) dt = \int_0^x p(t) dt + \int_0^x q(t) dt = T(p)(x) + T(q)(x)$ ✓ $T(cp)(x) = \int_0^x cp(t) dt = c\int_0^x p(t) dt = c T(p)(x)$ ✓

So $T$ is linear. Q7: Determining injectivity

Is $T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x,y) = (x, 0)$ injective?

Solution: $T(1,2) = (1,0)$ and $T(1,3) = (1,0)$. Different inputs give same output, so not injective. $\ker(T) = \{(0,y) \mid y \in \mathbb{R}\} \neq \{\mathbf{0}\}$. Q8: Determining surjectivity

Is $T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x,y) = (x, 0)$ surjective?

Solution: $C(A) = \text{span}\{(1,0)\} \neq \mathbb{R}^2$. For example, $(0,1)$ has no preimage. Not surjective. Q9: Non-linear transformation

Show $T: \mathbb{R}^2 \to \mathbb{R}$, $T(x,y) = x^2 + y^2$ is not linear.

Solution: $T(1,0) = 1$, $T(2,0) = 4$, but $T(2,0) = T(2(1,0)) = 4 \neq 2T(1,0) = 2$. Fails homogeneity.

Alternatively: $T((1,0)+(1,0)) = T(2,0) = 4 \neq 1+1 = T(1,0) + T(1,0)$. Fails additivity. Q10: Finding T from basis images

If $T: \mathbb{R}^2 \to \mathbb{R}^2$ is linear, $T(1,0) = (2,1)$, $T(0,1) = (3,4)$, find $T(-1,5)$.

Solution: $(-1,5) = -1(1,0) + 5(0,1)$. $T(-1,5) = -1T(1,0) + 5T(0,1) = -1(2,1) + 5(3,4) = (-2,-1) + (15,20) = (13,19)$.

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🔗 Cross-References

• Next topic: Kernel & Image
• Week 6 (Matrix Representation): How to represent $T$ as a matrix relative to any basis
• Week 7 (Similarity): Two matrices represent the same $T$ in different bases
• BSMA1001 (Maths 1): Functions, composition
• BSCS3004 (Deep Learning): Layers of a neural network are compositions of linear transformations + non-linear activations Join Discord Previous5.1 Null Space & Column SpaceNext6.1 Kernel & Image