- #1
chadpip
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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.