Lesson 2: The Matrix of a Linear Transformation
Estimated time: 35-45 minutes
Learning Objectives
- Construct the standard matrix of a linear transformation from its action on basis vectors
- Use the standard matrix to compute images of any vector
- Relate composition of transformations to matrix multiplication
The Standard Matrix
If T : R^n → R^m is linear, then T is completely determined by what it does to the standard basis vectors e1, e2, ..., en. Why? Because any vector x = x1*e1 + x2*e2 + ... + xn*en, and by linearity T(x) = x1*T(e1) + x2*T(e2) + ... + xn*T(en).
Standard Matrix: The standard matrix of T is the m x n matrix whose columns are the images of the standard basis vectors:
Then T(x) = Ax for all x in R^n.
Constructing the Standard Matrix
Worked Example 1
Let T : R^2 → R^3 be defined by T(x, y) = (x + 2y, 3x, y - x).
Step 1: Find T(e1) = T(1, 0) = (1, 3, -1).
Step 2: Find T(e2) = T(0, 1) = (2, 0, 1).
Step 3: Form the matrix with these as columns:
[ 3 0 ]
[-1 1 ]
Verify: T(2, 3) = (2+6, 6, 3-2) = (8, 6, 1). And A*[2;3] = [1(2)+2(3); 3(2)+0(3); -1(2)+1(3)] = [8; 6; 1]. ✓
Worked Example 2: Rotation by 60 degrees
T rotates vectors in R^2 counterclockwise by 60 degrees.
T(e1) = T(1,0) = (cos60, sin60) = (1/2, sqrt(3)/2).
T(e2) = T(0,1) = (-sin60, cos60) = (-sqrt(3)/2, 1/2).
[ sqrt(3)/2 1/2 ]
When You Know T on Other Vectors
Sometimes T is described by its action on non-standard vectors. You can still find the standard matrix by expressing the standard basis in terms of the given vectors, then using linearity.
Worked Example 3
T : R^2 → R^2 is linear with T(1, 1) = (2, 3) and T(1, -1) = (0, 1).
Express e1 in terms of (1,1) and (1,-1):
e1 = (1, 0) = (1/2)(1, 1) + (1/2)(1, -1).
T(e1) = (1/2)T(1,1) + (1/2)T(1,-1) = (1/2)(2,3) + (1/2)(0,1) = (1, 3/2) + (0, 1/2) = (1, 2).
Express e2: e2 = (0, 1) = (1/2)(1, 1) - (1/2)(1, -1).
T(e2) = (1/2)(2,3) - (1/2)(0,1) = (1, 3/2) - (0, 1/2) = (1, 1).
Standard matrix: A = [1 1; 2 1].
Composition of Transformations
Composition as Matrix Multiplication: If T1 : R^n → R^m has matrix A and T2 : R^m → R^p has matrix B, then the composition T2 composed with T1 has matrix BA.
Note the order: T2(T1(x)) = B(Ax) = (BA)x. The matrix of the outer transformation goes on the left.
Worked Example 4
T1 = reflection across the x-axis: A1 = [1 0; 0 -1].
T2 = rotation by 90 degrees: A2 = [0 -1; 1 0].
Composition T2(T1(x)): A2 * A1 = [0 -1; 1 0] * [1 0; 0 -1] = [0 1; 1 0].
This is reflection across y = x! First reflecting across x-axis, then rotating 90 degrees, gives a reflection across the diagonal.
Order Matters
A1 * A2 is generally NOT equal to A2 * A1. The order of composition matters -- just as putting on socks then shoes is different from shoes then socks.
Check Your Understanding
1. Find the standard matrix of T(x, y) = (x - y, 2x + 3y).
2. T : R^3 → R^2 maps T(1,0,0)=(2,1), T(0,1,0)=(0,-1), T(0,0,1)=(3,4). What is A?
3. If A is the matrix for "rotate 90 degrees" and B is the matrix for "rotate 90 degrees" again, what is BA?
4. Why is the standard matrix always unique for a given linear transformation?
Key Takeaways
- The standard matrix A has columns T(e1), T(e2), ..., T(en)
- Once you know A, computing T(x) = Ax for any x is just matrix-vector multiplication
- Composition of linear transformations corresponds to matrix multiplication -- and order matters
- If T is described via non-standard vectors, express the standard basis in terms of those vectors and use linearity