Linear Algebra, Theory And Applications, 2012a

Matrices And Linear
Transformations
2.1 Matrices
You have now solved systems of equations by writing them in terms of an augmented matrix
and then doing row operations on this augmented matrix. It turns out that such rectangular
arrays of numbers are important from many other different points of view. Numbers are
also called scalars. In general, scalars are just elements of some field. However, in the first
part 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 5 2 2 3 8 4
7
⎞
⎠
6 −9 1 2
This matrix is a 3 × 4 matrix because there are three rows and four columns. ⎛ The ⎞ first
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rowis(1234), the second row is (5 2 8 7) and so forth. The first column is ⎝ 5 ⎠ . The
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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 up
right while the rows just lay there like rows made by a tractor in a plowed field. Elements of
the matrix are identified according to position in the matrix. For example, 8 is in position
2, 3 because it is in the second row and the third column. You might remember that you
always list the rows before the columns by using the phrase Rowman Catholic. The symbol,
(a ij ) refers to a matrix in which the i denotes the row and the j denotes the column. Using
this notation on the above matrix, a 23 =8,a 32 = −9,a 12 =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, consider
the following example.
⎛
⎞ ⎛
⎞
1 2 3 4 3 6 9 12
3 ⎝ 5 2 8 7 ⎠ = ⎝ 15 6 24 21 ⎠ .
6 −9 1 2 18 −27 3 6
The new matrix is obtained by multiplying every entry of the original matrix by the given
scalar. 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 the
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