1. The Matrix of a Linear Transformation (9/30/05)
We observed last time that every transformation T from Rn to Rm which is described by a matrix is linear.
The opposite is also true: Every linear transformation T can be completely described by a unique matrix A, which is called the standard matrix for T .

2. An example
Suppose T is a linear transformation from R2 to R3 which takes the vector e1= to the vector and takes the vector e2= to the vector Find a matrix description of T .
How can we generalize this?

3. Explicit description of the matrix A
If T is a linear transformation from Rn to Rm , and if for each j , T takes the vector ej = (0,0,…,1,..,0) (only 1 is in the j th slot) in Rn to a vector vj in Rm, then the unique matrix A which describes T is simply the matrix whose columns are v1, v2, …, vn .
That is, T ‘s action on the ej’s completely describes T .

4. One-to-one and Onto
A transformation T from Rn to Rm is called onto if every b in Rm is the image of at least one a in Rn. (Hence Rm is the range of T .)
T is called one-to-one if every b in Rm is the image of at most one a in Rn.
If T is linear, we can check whether it’s one-to-one simply by checking what a’s get sent to 0.

5. Connections to Independence and Spanning
If a linear transformation T from Rn to Rm is represented by the m by n matrix A, then
T is onto if and only if the columns of A
span Rm.
T is one-to-one if and only if the columns of A are linearly independent.

6. An Application & Assignment for Monday
Section 1.10 contains three applications of which we will do one: Linear difference equations. See pages
For Monday:
Read Section 1.9.
Do Practice and Exercises 1, 3, 5, 11, 13, 17, 19, 21, 23, 25, 27.
Read pages and do Exercise 9 on page 101.