Linear Transformations
A transformation is simply a function (or mapping) from to i.e. for any (input) vector in , is an (output) vector in . In linear algebra, we will mainly study a very special type of transformations called linear transformations. They are transformations that preserve vector addition and scaling i.e. is a linear transformation if for any vectors in and any real number , we have
, and
Therefore, given the standard basis for , for any vector in , it can be written as a linear combination of the standard basis as follows:
When we apply a linear transformation to it, using the fact that it preserves vector addition and scaling, we get
You can see that is the linear combination of with the same weight. In other words, the linear transformation is uniquely determined by .
In the following applet, we consider any linear transformation . As mentioned above, we can define by specifying and . You can change and freely, then click the "Go" button and see how the grid in the domain is "transformed" under . Also, you can define the vector by inputting the coordinates of its arrowhead. The column vector will be shown. Again, you can click the "Go" button to see the transformation from to visually.
The following are some questions that test your understanding on linear transformations:
Will a grid always be transformed into a grid by any linear transformation? Explain your answer briefly.
Can a zero vector be transformed to a non-zero vector by a linear transformation? Explain your answer briefly.
Which of the following is a linear transformation from to ? You can select more than one answer.
Let such that and . Find . (You should first try to compute for the answer without using the above applet.)
Let such that , and . Find .