Surjective and injective linear map

[Fernando Revilla]

I quote an unsolved question from MHF posted by user jackGee on February 3rd, 2013.

[Let T:V->W Be A Linear Transformation
Where V and W are vector spaces over a Field F
let a={v₁,v₂,...,v_n} be a basis for V and b={w₁,w₂,...,w_m} be a basis for W

a) Prove that T is surjective if and only if the columns of [T]_ba span Fⁿ
b) Prove that T is injective if and only if the columns of [T]_ba are linearly independent in Fⁿ

P.S. Of course, I meant in the title and instead of an.

 

[Fernando Revilla]

$(a)\;$ Hint: Use $[T(x)]_B=[T]_{BA}[x]_A=[C_1,\ldots,C_n][x]_A=x_1C_1+\ldots+x_nC_n$ and the standard isomorphism between $V$ and $\mathbb{F}^n$ given by $x\to [x]_A$.

$(b)\;$ There is a typo. It should be $\mathbb{F}^m$ instead of $\mathbb{F}^n$, otherwise does not make sense. Hint: use $\dim (\ker T)=n-\mbox{rank }[T]_{BA}$.