The Matrix of a Linear Transformation

The Matrix of a Linear Transformation

Finding the Matrix

We have seen how to find the matrix that changes from one basis to another. We have also seen how to find the matrix for a linear transformation from R^m to Rⁿ. Now we will show how to find the matrix of a general linear transformation when the bases are given.

Definition

Let L be a linear transformation from V to W and let

S = {v₁, ... ,v_m} and T = {w₁, ... ,w_n}

be bases for V and W respectively. Then the matrix A representing L with respect to the bases S and T has ij^th component that is the j^th coordinate of the vector L(v_i).

We start with the example when both bases are "standard". For this section, we will use the following conventions. We will use the letter "E" to denote the standard basis.

For P_n, we will call the basis

(P_n)_E = {1, t, t², ..., tⁿ}

the "standard" basis.

For M^mxn, we will call the basis

M_E^mxn = {A₁₁, A₁₂, ..., A_1n, A₂₁, ..., A_2n,... ..., A_m1, ..., A_mn}

the standard basis where A_ij is the matrix with the ij^th entry equal to 1 and all the rest are zero. With this convention, the standard basis for M^3x2 is given by

Example

Let L be the linear transformation from R² to P₂ defined by

L(x,y) = x + yt + (x + y)t²

Find the matrix representing L with respect to the standard bases.

Solution

Notice that

L(1,0) = 1 + t² = (1,0,1) L(0,1) = t + t² = (0,1,1)

hence the matrix is given by

Now we will proceed with a more complicated example.

Example

Let L be the linear transformation from R² to R² such that

L(x,y) = (x - 2y, y - 2x)

and let

S = {(2, 3), (1, 2)}

be a basis for R². Find the matrix for L that sends a vector from the S basis to the standard basis.

Solution

This involves two parts. The first is to find the matrix for L from the standard basis to the standard basis. This matrix is found by finding

L(1, 0) = (1, -2) and L(0,1) = (-2, 1)

The matrix is

Next we find the matrix from the S basis to the standard basis E. This matrix is

Now consider the diagram below

The matrix that we want is the composition of these two mappings. Remembering that composition of functions is written from right to left we get

Example

Let L be the linear transformation from P₂ to P₂ with such that

L(a + bt + ct²) = (a + c) + (a + 2b)t + (a + b + 3c)t²

and let

S = {1 - t, 1 - t², t - t²) and T = {2 + t + t², 1 + t, 1 + t + t²}

Find the matrix of L with respect to the bases S and T.

Solution

We first find the matrix for L from the standard basis to the standard basis. We have

L(1,0,0) = L(1 + 0t + 0t²) = 1 + t + t² = (1,1,1)

L(0,1,0) = L(0 + t + 0t²) = 2t + t² = (0,2,1)

L(0,0,1) = L(0 + 0t + t²) = 1 + 3t² = (1,0,3)

Hence the matrix for L with the standard bases is

Next we find the transition matrices

Now we consider the diagram below

Since matrices are functions, we compose the functions by multiplying the matrices from right to left. We get

Example

Let L be the linear transformation from M^2x2 to M^2x2 and let

and

Find the matrix for L from S to S.

Solution

First we find the matrix for L in the standard basis. We have

so that

Next we have

We use the diagram

and our matrix is

Diagonalizing

The last example showed us that the matrix for L was of the form

P^-1AP

This was the definition of a matrix that is similar to A. If A is an n x n matrix and L is the linear transformation

L(v) = Av

and if the eigenvectors {v₁, ... ,v_n}are linearly independent then they form a basis for Rⁿ. With

S = {v₁, ... ,v_n}

then we have seen that if P is the matrix whose i^th column is v_i, then

D = P^-1AP

is a diagonal matrix. But this P is the transition matrix from the standard basis to the basis S. We have proven the following theorem.

Theorem

Let L be the linear transformation

L(v) = Av

then A is diagonalizable with n linearly independent eigenvectors S = {v₁, ... ,v_n} if and only if the matrix of L with respect to S is diagonal.