Page 121 - 35Linear Algebra

7

Matrices

Matrices are a powerful tool for calculations involving linear transformations.
It is important to understand how to find the matrix of a linear transforma-
tion and the properties of matrices.

7.1 Linear Transformations and Matrices

Ordered, finite-dimensional, bases for vector spaces allows us to express linear
operators as matrices.

7.1.1 Basis Notation

A basis allows us to efficiently label arbitrary vectors in terms of column
vectors. Here is an example.

Example 74 Let

a b
V = a, b, c, d ∈ R
c d
be the vector space of 2 × 2 real matrices, with addition and scalar multiplication
defined componentwise. One choice of basis is the ordered set (or list) of matrices

1 0 0 1 0 0 0 0 1 1 2 2
B = , , , =: (e , e , e , e ) .
1
2
1
2
0 0 0 0 1 0 0 1
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