Page 134 - 35Linear Algebra

134 Matrices

i
The numbers m are called entries. The superscript indexes the row of
j
the matrix and the subscript indexes the column of the matrix in which m i j
appears.

r
r
An r × 1 matrix v = (v ) = (v ) is called a column vector, written
1
 
v 1
 v 2 
v =  .  .


.
 . 
v r
1
A 1 × k matrix v = (v ) = (v k ) is called a row vector, written
k

v = v 1 v 2 · · · v k .

The transpose of a column vector is the corresponding row vector and vice
versa:

Example 81 Let
 
1
v =   .
2
3
Then

T
v = 1 2 3 ,
T T
and (v ) = v. This is an example of an involution, namely an operation which when
performed twice does nothing.

A matrix is an efficient way to store information.

Example 82 In computer graphics, you may have encountered image files with a .gif
extension. These files are actually just matrices: at the start of the file the size of the
matrix is given, after which each number is a matrix entry indicating the color of a
particular pixel in the image.
This matrix then has its rows shuffled a bit: by listing, say, every eighth row, a web
browser downloading the file can start displaying an incomplete version of the picture
before the download is complete.
Finally, a compression algorithm is applied to the matrix to reduce the file size.

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