The linear transformation T:Rn→Rm is defined by T(v)=Av.Find:-

Question

the dimensions of Rⁿ and R^m.

1-A= ⌈ 0 1 -2 1⌉

-1 4 5 0

⌊ 0 1 3 1⌋.

2-A= ⌈ 1 2 ⌉

-2 4

⌊-2 2 ⌋.

Answer 1

The key to answering this question is to recall how a matrix transforms the dimension of a vector and why they do what they do.

When left multiplying a vector by a matrix, the dimension of the vector has to match the number of columns in the matrix. The result, however, will be a vector with a dimension equal to the number of rows in the matrix.

For the first question, A is a 3x4 matrix. That means T maps R⁴ → R³.

With this information, I think you can do the second one yourself.