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# Finding a matrix for a linear transformation. Watch

1. Consider the subset

of and the function

given by

1/ Find a basis of X
2/ Find the matrix for f wrt this basis.

I got a basis of (1,1,0,3),(1,0,1,0)

But I'm getting really confused about finding the matrix because the vector space only has 2 vectors in the basis but the vectors themselves have 4 entries. Can someone explain how to do this?
2. (Original post by TheJ0ker)
Consider the subset

of and the function

given by

1/ Find a basis of X
2/ Find the matrix for f wrt this basis.

I got a basis of (1,1,0,3),(1,0,-1,0)

But I'm getting really confused about finding the matrix because the vector space only has 2 vectors in the basis but the vectors themselves have 4 entries. Can someone explain how to do this?
If e1,e2 are your two basis vectors, then any vector can be written in the form ae1+be2

Then you are looking to find a 2x2 matrix that takes (a,b)^T and transforms it into the appropriate new vector ce1+de2

So you need to find what f does to e1, e2, and express the result in terms of e1 and e2, in each case.
3. (Original post by ghostwalker)

If e1,e2 are your two basis vectors, then any vector can be written in the form ae1+be2

Then you are looking to find a 2x2 matrix that takes (a,b)^T and transforms it into the appropriate new vector ce1+de2

So you need to find what f does to e1, e2, and express the result in terms of e1 and e2, in each case.
Yeah sorry, I meant (1,0,1,0);

so let e1 = (1,1,0,3), e2 = (1,0,1,0)

then f(e1) = (1,1,0,3) = (1)e1 + 0e2
and f(e2) = (1,1,0,3) = (1)e1 + 0e2

then how would I find the matrix? Is it
?

And why is it a 2x2 surely I need a 4x4 if I want to apply the matrix transformation to a vector in X?
4. It's not that matrix actually is it
5. (Original post by TheJ0ker)
Yeah sorry, I meant (1,0,1,0);

so let e1 = (1,1,0,3), e2 = (1,0,1,0)

then f(e1) = (1,1,0,3) = (1)e1 + 0e2
and f(e2) = (1,1,0,3) = (1)e1 + 0e2

then how would I find the matrix? Is it
?
Close. You need to transpose that.

And why is it a 2x2 surely I need a 4x4 if I want to apply the matrix transformation to a vector in X?
Your matrix is relative to a basis, in this case your chosen one.
The domain is two dimensional, any vector is represented in the form ae1+be2, relative to that basis. I.e. there are only two co-ordinates.
6. (Original post by ghostwalker)
Close. You need to transpose that.

Your matrix is relative to a basis, in this case your chosen one.
The domain is two dimensional, any vector is represented in the form ae1+be2, relative to that basis. I.e. there are only two co-ordinates.
Ah thank you so so much, I've been getting so frustrated over this all day and I understand the whole thing now.
7. (Original post by TheJ0ker)
Ah thank you so so much, I've been getting so frustrated over this all day and I understand the whole thing now.
8. Isn't (1,0) , (0,1) a basis here given that everything is basically a linear combination of these two?
9. (Original post by Zilch)
Isn't (1,0) , (0,1) a basis here given that everything is basically a linear combination of these two?

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