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 Rm 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, ... ,vm}         and         T  = {w1, ... ,wn

be bases for V and W respectively.  Then the matrix A representing L with respect to the bases S and T has ijth component that is the jth coordinate of the vector L(vi).

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 Pn, we will call the basis

        (Pn)E  =  {1, t, t2, ..., tn}

the "standard" basis.

For Mmxn, we will call the basis

      MEmxn  =  {A11, A12, ..., A1n, A21, ..., A2n,... ..., Am1, ..., Amn}

the standard basis where Aij is the matrix with the ijth entry equal to 1 and all the rest are zero.  With this convention, the standard basis for M3x2 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 M2x2 to M2x2 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, ... ,vn}are linearly independent then they form a basis for Rn.  With

        S  =  {v1, ... ,vn}

then we have seen that if P is the matrix whose ith column is vi, 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, ... ,vn} if and only if the matrix of L with respect to S is diagonal.


Orthogonal Matrices

We finish this discussion with a little geometry.  

A linear transformation L called an isometry if for any u and v in V,

        L(u) . L(v)  =  u . v

Notice that isometries preserve angles.  


Theorem

L is an isometry if and only if the matrix of L with respect to the natural basis is orthogonal.

 

Proof

We use the fact that for any vectors x and y and matrix A,

        (Ax) . y  =  x . (ATy)

If L is an isometry, then 

        u . v  =  L(u) . L(v)  =  Au . Av  =  u . (ATAv)

for all u and v.  But then 

        v  =  ATAv 

so that 

        ATA  =  I

that is, A is orthogonal

Conversely, if A is orthogonal, then

        L(u) . L(v)  =  Au . Av  =  u . (ATAv)  =  u . v 

so that L is an isometry.



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