Finding the Matrix of a Linear Transformation

[bakin]

Homework Statement

Assume that T defines a linear transformation and use the given information to find the matrix of T
T: R⁴-->R²
such that T(1,0,0,0)=(3,-2), T(1,1,0,0)=(5,1), T(1,1,1,0)=(-1,0), and T(1,1,1,1)=(2,2)

Homework Equations

The Attempt at a Solution

I think I need to use/find the standard basis for R⁴, but not really sure.

Thanks for the help.

 

[phreak]

The standard basis for R^4 is (1,0,0,0), (0,1,0,0), (0,0,1,0), and (0,0,0,1). The idea of this problem is that once you know where the basis elements go, you can uniquely determine what the matrix is (using methods described in your textbook!)

So you need to find T(1,0,0,0), which you already know, along with T(0,1,0,0), T(0,0,1,0), and T(0,0,0,1). How would you find these? (Hint: Use the definition of linearity.)

 

[bakin]

Do I set them as linear combinations of each other? [a b c d]=c1v1+c2v2+c3v3+c4v4 ?

 

[bakin]

OIWQJFEOIWJQEF just figured it out.

for example, in order to find (0,1,0,0), we can say that's

(0,1,0,0) = a(1,1,0,0)-b(1,0,0,0) if we set a=b=1

T(0,1,0,0)=T(1,1,0,0)-T(1,0,0,0)
T(0,1,0,0)=(5,1)-(3,-2)
T(0,1,0,0)=(2,3) which is indeed the answer provided. I just have to do this for the rest of the vectors.thanks for the help