Matrix of a Linear Map
If in bases of and , then with .
Consider a linear map , with and , and take two bases of and of .
Then, can be represented by , which is an array with entries for as the rows and as the columns, such that
for . This automatically ensures that for any , , we can always write and in terms of the bases:
This means that any coefficient from the image can be written as
To summarise, given and which are real or complex vector spaces with and , and given bases of and of , then
- is identified with or .
- is identified with or .
- We identify the linear map with the matrix such that .
Linear combinations of matrices
If has matrix representation with respect to the same bases, then for scalars the map is represented by the matrix with coefficients
because addition and scalar multiplication of matrices take place entry-wise.
Products represent compositions
If is represented by the matrix and is represented by , then the composition is represented by the matrix product with coefficients
the dot product of row of with column of . The number of columns of must equal the number of rows of for the product to be defined, and has the same number of rows as and the same number of columns as . For all matrices for which the products are defined, and scalars :
Example
Consider and . Hence and . Consider the map
The map is linear. We want to find the matrix representation of with respect to the bases
of and
of .
To determine , we need to compute for :
Therefore, for ,
Thus, the matrix representation of with respect to the given bases is
Related
Stated in
- Definition 3.9 (Matrix)§3.4.1 Definitions
- Example 3.10§3.4.1 Definitions