Linear Algebra, advanced mathematics

Question

Okay, let e₁ = (1,0), e₂ = (0,1), y₁ = (2,5), and y₂=(-1,6). Let T: R² →R² be a transformation that maps e₁ into y₁ and maps e₂ into y₂. Find the image of (5, -3) under T. Is T onto? Is it One-to-one? Justify.

Answer 1

The answer is simply T[(5,-3)]=5T[(1,0)]-3T[(0,1)]=5(2,5)-3(-1,6)=(10,25)+(3,-18)=(13,7). Also, since the basis e_{1}, e_{2} is mapped onto another basis of R^{2} namely, (2,5) and (-1,6) we have that the linear transformation T is 1-1 and onto R^{2}.