Matrix Representation of Linear Transformations

🎯 Learning Objectives

• Find the matrix of a linear transformation relative to given ordered bases
• Compute the image of a vector using the matrix representation
• Understand how the matrix changes when bases change
• Compute change-of-basis matrices
• Determine transition matrices between bases

📋 Prerequisites

• Linear Transformations (Week 5) — definition and properties
• Basis & Dimension (Week 4) — coordinates relative to a basis
• Matrix Multiplication (Week 1) — matrices as linear maps

1. Intuition: Translating Between Languages

• Represent any vector $\mathbf{v} \in V$ by its coordinate vector $[\mathbf{v}]_B \in \mathbb{R}^n$
• Represent $T$ by a matrix $M$ so that $[T(\mathbf{v})]_C = M [\mathbf{v}]_B$ (Diagram)

2. Coordinates Relative to a Basis

2.1 Definition

Example 1: Coordinates

$B = \{1-x, 1+x, x^2\}$ is a basis for $P_2$.

For $p(x) = 3 + x + 2x^2$, find $[p]_B$.

Solve $c_1(1-x) + c_2(1+x) + c_3 x^2 = 3 + x + 2x^2$. Group: $(c_1 + c_2) + (-c_1 + c_2)x + c_3 x^2 = 3 + x + 2x^2$.

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3. Matrix of a Linear Transformation

Definition (Matrix of $T$). Let $T: V \to W$ be linear, $B = \{\mathbf{b}_1, \dots, \mathbf{b}_n\}$ an ordered basis of $V$, and $C = \{\mathbf{c}_1, \dots, \mathbf{c}_m\}$ an ordered basis of $W$. The matrix of $T$ relative to $B$ and $C$ is:

It satisfies: $[T(\mathbf{v})]_C = [T]_{C \leftarrow B} [\mathbf{v}]_B$.

3.1 Algorithm

1. Compute $T(\mathbf{b}_j)$ for each basis vector $\mathbf{b}_j$ of $V$
2. Express $T(\mathbf{b}_j)$ in the basis $C$ of $W$ to get $[T(\mathbf{b}_j)]_C$
3. These coordinate vectors become the columns of $[T]_{C \leftarrow B}$

Example 2: Matrix of a transformation

$T: P_2 \to P_1$, $T(p) = p'$ (derivative). $B = \{1, x, x^2\}$ (basis for $P_2$) $C = \{1, x\}$ (basis for $P_1$)

Check: For $p(x) = a + bx + cx^2$, $p'(x) = b + 2cx$. $[p]_B = (a,b,c)^T$.

= $[p']_C$ ✓ Example 3: Standard matrix = matrix relative to standard bases

$T: \mathbb{R}^3 \to \mathbb{R}^2$, $T(x,y,z) = (2x-y, x+3y+z)$. Using standard bases $B = \{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\}$, $C = \{\mathbf{e}_1,\mathbf{e}_2\}$:

(the standard matrix we found in Week 5).

4. Change of Basis

4.1 Change-of-Basis Matrix

Definition (Change-of-Basis Matrix). The change-of-basis matrix from $B$ to $B'$ is:

It satisfies: $[\mathbf{v}]_{B'} = P_{B' \leftarrow B} [\mathbf{v}]_B$. Key fact: $P_{B' \leftarrow B} = (P_{B \leftarrow B'})^{-1}$. Example 4: Change of basis in ℝ²

Standard basis $S = \{\mathbf{e}_1, \mathbf{e}_2\}$, new basis $B = \{(1,2), (3,4)\}$. Find $P_{B \leftarrow S}$ (from standard to $B$).

$\mathbf{e}_1 = (1,0)$ in $B$: solve $c_1(1,2) + c_2(3,4) = (1,0)$.

So

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Similarly, $\mathbf{e}_2 = (0,1)$ in $B$: solve $c_1(1,2) + c_2(3,4) = (0,1)$. $c_1 = \frac{3}{2}, c_2 = -\frac{1}{2}$.

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4.2 Effect on the Matrix of a Transformation

5. Edge Cases & Gotchas

6. Common Pitfalls

❌ Pitfall 1: Wrong order in $[T]_{C \leftarrow B}$

❌ Pitfall 2: Confusing change-of-basis direction

❌ Pitfall 3: Forgetting to convert the output basis

7. Formula Summary Table

8. 📝 Practice Questions

Q1: Matrix relative to non-standard bases

$T: \mathbb{R}^2 \to \mathbb{R}^2$, $T(x,y) = (x+2y, 3x+4y)$. $B = \{(1,1), (1,-1)\}$, $C = \{(1,0), (0,1)\}$. Find $[T]_{C \leftarrow B}$.

Solution: $T(1,1) = (3,7)$, $[T(1,1)]_C = (3,7)$ (standard basis). $T(1,-1) = (-1,-1)$, $[T(1,-1)]_C = (-1,-1)$.

. Q2: Matrix of a derivative operator with different bases

$T: P_2 \to P_1$, $T(p) = p'$. $B = \{1, 1+x, x+x^2\}$, $C = \{1, x\}$. Find $[T]_{C \leftarrow B}$.

Solution: $T(1) = 0$, $[T(1)]_C = (0,0)$. $T(1+x) = 1$, $[T(1+x)]_C = (1,0)$. $T(x+x^2) = 1+2x$, $[T(x+x^2)]_C = (1,2)$.

. Q3: Change of basis

Find $P_{B \leftarrow S}$ where $B = \{(1,0,0), (1,1,0), (1,1,1)\}$, $S$ = standard basis.

Solution: $\mathbf{e}_1 = 1\cdot(1,0,0)$, so $[\mathbf{e}_1]_B = (1,0,0)$. $\mathbf{e}_2 = -(1,0,0) + 1\cdot(1,1,0)$, so $[\mathbf{e}_2]_B = (-1,1,0)$. $\mathbf{e}_3 = 0\cdot(1,0,0) + (-1)(1,1,0) + 1\cdot(1,1,1)$, so $[\mathbf{e}_3]_B = (0,-1,1)$.

. Q4: Using the matrix to compute

Using $[T]_{C \leftarrow B}$ from Q1, find $T(2,3)$.

Solution: Find $[(2,3)]_B$: solve $c_1(1,1)+c_2(1,-1) = (2,3)$ ⇒ $c_1+c_2=2$, $c_1-c_2=3$ ⇒ $c_1=2.5$, $c_2=-0.5$.

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$T(2,3) = (8,18)$ (in standard coordinates). Check directly: $T(2,3) = (2+6, 6+12) = (8,18)$ ✓ Q5: Similarity transformation

$T: \mathbb{R}^2 \to \mathbb{R}^2$ has matrix

in standard basis. Find $[T]_B$ for $B = \{(1,1), (1,-1)\}$.

Solution: $[T]_B = P^{-1} A P$ where $P = P_{B \leftarrow S}$.

(columns are basis vectors).

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. Q6: Finding transition matrix

$B = \{(1,0), (1,1)\}$, $B' = \{(2,1), (1,2)\}$. Find $P_{B' \leftarrow B}$.

Solution: For each vector in $B$, find coordinates in $B'$.

$(1,0) = c_1(2,1) + c_2(1,2)$: $2c_1 + c_2 = 1$, $c_1 + 2c_2 = 0$ ⇒ $c_2 = -\frac{1}{3}$, $c_1 = \frac{2}{3}$.

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$(1,1) = c_1(2,1) + c_2(1,2)$: $2c_1 + c_2 = 1$, $c_1 + 2c_2 = 1$ ⇒ $c_1 = \frac{1}{3}$, $c_2 = \frac{1}{3}$.

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. Q7: Matrix of identity

What is $[I]_{C \leftarrow B}$ where $I: V \to V$ is the identity map?

Solution: $I(\mathbf{b}_j) = \mathbf{b}_j$. So $[I]_{C \leftarrow B}$ has columns $[\mathbf{b}_j]_C$. This is exactly the change-of-basis matrix $P_{C \leftarrow B}$! Q8: Composition

$T: V \to W$, $S: W \to U$ are linear. Prove $[S \circ T]_{D \leftarrow B} = [S]_{D \leftarrow C} [T]_{C \leftarrow B}$.

Proof: For any $\mathbf{v} \in V$: $[(S \circ T)(\mathbf{v})]_D = [S(T(\mathbf{v}))]_D = [S]_{D \leftarrow C} [T(\mathbf{v})]_C = [S]_{D \leftarrow C} [T]_{C \leftarrow B} [\mathbf{v}]_B$.

So $[S \circ T]_{D \leftarrow B} = [S]_{D \leftarrow C} [T]_{C \leftarrow B}$. ✓ Q9: Matrix of a projection

$T: \mathbb{R}^3 \to \mathbb{R}^3$, $T(x,y,z) = (x,y,0)$. Find $[T]_B$ where $B = \{(1,0,0), (1,1,0), (1,1,1)\}$.

Solution: $T(1,0,0) = (1,0,0)$. In $B$: solve $(1,0,0) = c_1(1,0,0)+c_2(1,1,0)+c_3(1,1,1)$ ⇒ $c_1=1$, $c_2=0$, $c_3=0$. $[T(1,0,0)]_B = (1,0,0)$.

$T(1,1,0) = (1,1,0)$ = $0(1,0,0)+1(1,1,0)+0(1,1,1)$. $[(1,1,0)]_B = (0,1,0)$.

$T(1,1,1) = (1,1,0)$ = $0(1,0,0)+1(1,1,0)+0(1,1,1)$. $[(1,1,1)]_B = (0,1,0)$.

. Q10: Going back and forth

If

and

, find $[T(\mathbf{v})]_C$.

Solution:

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🔗 Cross-References

• Next topic: Equivalent & Similar Matrices
• Week 7 (Similarity): $A$ and $P^{-1}AP$ represent the same operator in different bases
• Week 8 (Orthogonal Transformations): Orthogonal matrices represent rotations/reflections
• BSCS3004 (Deep Learning): Weight matrices in neural networks are linear transformations between feature spaces Join Discord Previous6.1 Kernel & ImageNext7.1 Equivalent & Similar Matrices