Lin algebra: find the matrix with respect to basis

[chadpip]

Homework Statement

Let W be a 4dim vector space with basis {e1, e2, e3, e4}. Let T be the linear mapping:

T(e1) = -e1 -2e2 + 2e3
T(e2) = 4e1 + 4e2 - 5e3 -3e4
T(e3) = 2e1 + 2e2 -3e3 -2e4
T(e4) = -e2 + e3

Let V be the subspace spanned by {e1 + e2 - e3, e1 - e4, -e1 + e2 -e3 +2e4}

Now: find a basis for V and calculate the matrix T with respect to V (the matrix T restricted to the subspace V) with respect to this basis

Homework Equations

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The Attempt at a Solution

well the 3 elements in the span of V are lin. DEP and i found that {e1 + e2 - e3, e1 - e4} are lin ind so they form a basis for V.

Now, for the matrix.. I keep getting confused. It seems that I am beginning in R4 and ending in R2... i am confused on how to get from a 4x4 matrix 2x4? I calcluated what the basis vectors for V look like when they go through the transformation T (** I got: T(e1 + e2 - e3) = e1 - e4 ; T(e1 - e4) = - e1 - e2 + e3).

I was thinking I would multiply the matrix rep. of T by something to give me the matrix rep. of **. Is this correct for what I should be doing?

But it doesn't really make sense...

I know my final answer needs to be a square matrix because later parts of this exercise ask me to calculate the eigenvalues (so it must be square.)

Hopefully I did not really confuse you. All the examples I have been looking at to try and help don't seem to be completely relevant.

 

[matt grime]

You've found a basis of V. Now, does T map V into itself? If so it is an linear map on T. What is it?

 

[chadpip]

since T is linear,
T(V) =
T(e1 - e4) = T(e1) - T(e4) = -e1 -e2 + e3
T(e1 + e2 - e3) = T(e1) -T(e2) + T(e3) = e1 - e4


the results come out of elements of the span, or linear combinations of elements of the span of V.

therefore, T(V) is contained in V, so T maps V into itself..

 

[matt grime]

So if we let f_1 and f_2 stand for the basis vectors e_1 -e_4 and e_1+e_2-e_3. T sends f_1 to what (in terms of f_i)?

 

[chadpip]

ah okay. then:

T(f_1) = - f_2
and
T(f_2) = f_1

.i feel we are now ready to construct the matrix knowing this but i am not sure what it is yet. I am thinking

 

[chadpip]

would then matrix then just be:

0 -1
1 0

?

 

[matt grime]

You shouldn't need to ask; it is a matter of verifying if it does what it is supposed to. It is the standard 'plugging the answer back in' to check.