Chapter 2

Linear Transformations

2.1 Matrices

You have now solved systems of equations by writing them in terms of an augmented matrixand then doing row operations on this augmented matrix. It turns out that such rectangulararrays of numbers are important from many other different points of view. Numbers arealso called scalars. In general, scalars are just elements of some field. However, in the firstpart of this book, the field will typically be either the real numbers or the complex numbers.

A matrix is a rectangular array of numbers. Several of them are referred to as matrices.For example, here is a matrix.  1 2 3 4

5 2 8 7

6 −9 1 2

This matrix is a 3 × 4 matrix because there are three rows and four columns. The first

row is (1 2 3 4) , the second row is (5 2 8 7) and so forth. The first column is

 1

5

6

 . The

convention in dealing with matrices is to always list the rows first and then the columns.Also, you can remember the columns are like columns in a Greek temple. They stand upright while the rows just lie there like rows made by a tractor in a plowed field. Elements ofthe matrix are identified according to position in the matrix. For example, 8 is in position2, 3 because it is in the second row and the third column. You might remember that youalways list the rows before the columns by using the phrase Rowman Catholic. The symbol,(aij) refers to a matrix in which the i denotes the row and the j denotes the column. Usingthis notation on the above matrix, a23 = 8, a32 = −9, a12 = 2, etc.

There are various operations which are done on matrices. They can sometimes be added,multiplied by a scalar and sometimes multiplied. To illustrate scalar multiplication, considerthe following example.

3

 1 2 3 4

5 2 8 7

6 −9 1 2

 =

 3 6 9 12

15 6 24 21

18 −27 3 6

 .

The new matrix is obtained by multiplying every entry of the original matrix by the givenscalar. If A is an m× n matrix −A is defined to equal (−1)A.

Two matrices which are the same size can be added. When this is done, the result is thematrix which is obtained by adding corresponding entries. Thus 1 2

3 4

5 2

+

 −1 4

2 8

6 −4

 =

 0 6

5 12

11 −2

 .

Two matrices are equal exactly when they are the same size and the corresponding entriesare identical. Thus  0 0

0 0

0 0

 ̸=

(0 0

0 0

)

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