Question 8159: is there any linear transformation from R^3 to R^2 defined by
T(1,-1,1)=(1,0)
T(1,1,1)=(0,1)
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Yes, since (1,-1,1) and (1,1,1) are linearly independent in R^3
Also, (1,0) and (0,1) are linearly independent in R^2 can form a basis.
We know that any linear transformation is uniquely determined by
the the values on the basis.
Set another vector v in R^3, which is independent of (1,-1,1) and (1,1,1),
say (0, 0,1) , then define T(0,0,1)= (0,0) [or choose v = (1,-1,1)x(1,1,1)
We obtain a linear transformation from R^3 to R^2 generated by
T(1,-1,1)=(1,0)
T(1,1,1)=(0,1) with Kernel(T) = <(0,0,1)>
More precisely, T(a(1,-1,1)+b(1,1,1)+c(0,0,1))
= a(1,0)+ b(0,1)= (a,b) for all real a,b,c
Kenny
PS: Important notice:
I will not solve your or other student's questions with repeated posting.
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