# Describing explicitly a linear transformation

• Feb 22nd 2010, 09:04 AM
Zalren
Describing explicitly a linear transformation
Describe explicitly a linear transformation T : R3 -> R3 which has as it ranges the subspace spanned by (1, 0, -1) and (1, 2, 2).

This is what I wrote:

T(R3) = a(1, 0, -1) + b(1, 2, 2) for all a, b in K (the underlying field).

Since (1, 0, -1) and (1, 2, 2) span the range, I assume that means that you can take the linear combination of those two vectors and generate the range. But not sure if I did it right.
• Feb 23rd 2010, 05:14 AM
HallsofIvy
Quote:

Originally Posted by Zalren
Describe explicitly a linear transformation T : R3 -> R3 which has as it ranges the subspace spanned by (1, 0, -1) and (1, 2, 2).

This is what I wrote:

T(R3) = a(1, 0, -1) + b(1, 2, 2) for all a, b in K (the underlying field).

Since (1, 0, -1) and (1, 2, 2) span the range, I assume that means that you can take the linear combination of those two vectors and generate the range. But not sure if I did it right.

I'm not sure what you mean by "describe explicitely". Yes, any vector in the range of T can be written as a(1, 0, -1)+ b(1, 2, 2) and so T(v) (I would not write "T(R3)") must be of that form for some a and b.

If you have no further information on T, that's about all you can say.