Practice question · Multiple choice
A linear map on the plane is completely determined once you know what it does to (1,0) and (0,1) - two vectors out of infinitely many. Why do those two pin down every other value?
Hints
- Write (4,7) in terms of (1,0) and (0,1), then apply T and use linearity.
- Which property of T lets you push it through a sum and a scalar multiple?
Show the answer
D. Because every vector is a combination of them, and linearity respects that.
Why
Write the vector in the basis and let linearity do the rest: (4,7) = 4(1,0) + 7(0,1), so two known outputs determine the third. This is what makes matrices possible, storing the images of the basis vectors as columns captures an infinite map in finitely many numbers. The two must be independent: T(1,0) and T(2,0) tell you almost nothing.
Practise Linear Transformations
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