The Matrix of a Linear Transformation

These notes closely follow the presentation of the material given in David C. Lay’s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation and should not be regarded as a substitute for thoroughly reading the textbook itself and working through the exercises therein.

The Matrix of a Linear Transformation We have seen that any matrix transformation x Ax is a linear transformation. The converse is also true. Specifically, if T :  n   m is a linear transformation, then there is a unique m  n matrix, A, such that Tx  Ax for all x  n . The next example illustrates how to find this matrix. Example Let T :  2   3 be the linear transformation defined by x 1  2x 2

x1

T



x2

.

x 2 3x 1  5x 2

Find the matrix, A, such that Tx  Ax for all x  2 . Solution The key here is to use the two “standard basis” vectors for  2 . These are the vectors e1 

Any vector x  x

x1 x2 x1 x2

1

0

and e 2 

0

1

.

  2 is a linear combination of e 1 and e 2 because x1



0



0

 x1e1  x2e2.

x2

Since T is a linear transformation, we know that Tx  Tx 1 e 1  x 2 e 2   x 1 Te 1   x 2 Te 2  

x1

Te 1  Te 2 

x2

 Ax where A is the 3  2 matrix A 

Te 1  Te 2 

.

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Since Te 1   T

Te 2   T

1 0

0 1

1  20 

1 

0



0

31  50

3

0  21

2 

1

1

,

5

30  51

we see that 1 2 A

0 1

.

3 5

In general, if T :  n   m is a linear transformation and A

Te 1  Te 2   Te n 

where 1 e1 

0  0

0 , e2 

1  0

0 , , e n 

0  1

is the set of standard basis vectors for  n , then Tx  Ax for all x  n .

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Example Find the linear transformation T :  2   2 that rotates each of the vectors e 1 and e 2 counterclockwise 90  . Then explain why T rotates all vectors in  2 counterclockwise 90  . Solution The T we are looking for must satisfy both Te 1   T

1 0



0 1

and Te 2   T

0 1



1 0

.

The standard matrix for T is thus A

0 1 1 0 x1

and we know that Tx  Ax for all x  2 . Hence, for any x 

x2

  2 , we

have Tx  Ax 

0 1

x1

1 0

x2



x 2 x1

.

By using some basic trigonometry, we can see that Tx is x rotated counterclockwise 90  . (See the figure below.)

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Example Find the linear transformation T :  2   2 that perpendicularly projects both of the vectors e 1 and e 2 onto the line x 2  x 1 . Then explain why T perpendicularly projects all vectors in  2 onto the line x 2  x 1 .

The T we are looking for must satisfy both Te 1   T

1 0



1 2 1 2



1 2 1 2

and Te 2   T

0 1

.

(See the figure below.)

The standard matrix for T is thus A

1 2 1 2

1 2 1 2

x1

and we know that Tx  Ax for all x  2 . Hence, for any x 

  2 , we

x2

have Tx  Ax 

1 2 1 2

1 2 1 2

x1 x2



1 2 1 2

x1  x1 

1 2 1 2

x2 x2

.

4

By using some basic trigonometry, we can see that Tx is the perpendicular projection of x onto the line x 2  x 1 . (See the figures below.)

5

“One–to–One” Linear Transformations and “Onto” Linear Transformations Definition A transformation T :  n   m is said to be onto  m if each vector b   m is the image of at least one vector x   n under T. Example The linear transformation T :  2   2 that rotates vectors counterclockwise 90  is onto  2 . Example The linear transformation T :  2   2 that perpendicularly projects vectors onto the line x 2  x 1 is not onto  2 . For example, there is no x   2 such that 1 . Tx  5 Definition A transformation T :  n   m is said to be one–to–one if each vector b   m is the image of at most one vector x   n under T. Example The linear transformation T :  2   2 that rotates vectors counterclockwise 90  is one–to–one. Example The linear transformation T :  2   2 that perpendicularly projects vectors onto the line x 2  x 1 is not one–to–one. For example, the vector

1 2 1 2

is the

image of both e 1 and e 2 under T.

6

Example Let T :  4   3 be the linear transformation whose standard matrix is 1 4 8 1 A

0 2 1 3 0 0

.

0 5

Does T map  4 onto  3 . Is T one–to–one?

Theorem Let T :  n   m be a linear transformation and let A be the standard matrix for T. Then T is one–to–one if and only if the homogeneous equation Ax  0 m has only the trivial solution. Theorem Let T :  n   m be a linear transformation and let A be the standard matrix for T. Then: 1. 2.

T maps  n onto  m if and only if the columns of A span  m . T is one–to–one if and only if the columns of A are linearly independent.

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