January 13th, 2009, 01:05 AM
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Quote: |
Look at the set of vectors {(1,0,1), (2,1,2), (0,1,1), (2,3,3)}. The first three of these vectors are linearly independent. Thus they form a basis for R^3, and you can express the vectors (1,0,0), (0,1,0), (0,0,1) in terms of them. Then use the linearity of T to find T(1,0,0), T(0,1,0) and T(0,0,1) in terms of the vector v=T(1,0,1). Finally, check whether v=T(2,3,3) is equal to 2*T(1,0,0) + 3*T(0,1,0) + 3*T(0,0,1). If so (and assuming that v is not the zero vector) then you have constructed a nonzero T with the required properties.
| How do I find the T(1,0,0), T(0,1,0) and T(0,0,1) vectors in terms of only T(1,0,1)??? |