Linear algebra: f:V->V invertible; prove {f(v_1), f(v_2), f(v_3)} lin. indep. set

[Tellei]

Consider a vector space V of dimension 3 and f a linear transformation from V to V such that f is invertible; that is, Ker(f ) = {(0, 0, 0)}. Let {v₁, v₂, v₃} be a set of linearly independent vectors of V.

(a) Prove that {f (v₁), f (v₂), f (v₃)} is a set of linearly independent vectors.

(b) Is {f (v₁), f (v₂), f (v₃)} a basis of V? Justify your answer.

Really struggling with this question. It's a further maths exercise and I'm really intrigued in a solution

 

[pka]

Consider a vector space V of dimension 3 and f a linear transformation from V to V such that f is invertible; that is, Ker(f ) = {(0, 0, 0)}. Let {v₁, v₂, v₃} be a set of linearly independent vectors of V.

(a) Prove that {f (v₁), f (v₂), f (v₃)} is a set of linearly independent vectors.

(b) Is {f (v₁), f (v₂), f (v₃)} a basis of V? Justify your answer.

Really struggling with this question. It's a further maths exercise and I'm really intrigued in a solution

Do you know what it means for a collection of vectors is linearly independent?
If so, assume the image is not independent. Then prove this by contradiction.
Using the properties of linear transformations show that that implies that the pre-image set must be dependent.

 

[HallsofIvy]

Of course, once you have proved (a), (b) is easy. A "basis" for a n dimensional vector space has three properties:
1) the vectors in the basis are independent.
2) they span the space.
3) there are n vectors in the basis.

And any two of those is sufficient to prove the third!