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Homework Statement
Find the linear map [tex]f:R^2 \rightarrow R^3[/tex], with f(1,2) = (2,1,0) and f(2,1)=(0,1,2)
Homework Equations
The Attempt at a Solution
I actually don't understand this task. PLease help! Thank you...
Excellent![tex]a=-\frac{x-2y}{3}[/tex]
[tex]b=\frac{2x-y}{3}[/tex]
,
now what to do next?
Haven't we already been through what a "basis" is? R^{2} has dimension 2 and, since (1, 2) and (2, 1) are two independent vectors (one is not a multiple of the other) they form a basis. Every vector in can be written as a linear combination of those two vectors. You have now determined how they can be written: for any x, y,Actually, I don't know why you multiply by [1 2] and why it is equal to [2 1 0]
Because they were the only ones ones for which we knew what f does!Ok, I understand now how you find it. But I can't understand what we do actually to find it.
f(x,y)= af(1,2)+ bf(2,1)
f(x,y)= a(2,1,0)+ b(0,1,2)[/quote
we actually make linear combination of the 2 vectors in the basis, why?