Surjective and injective linear map

Fernando Revilla

Well-known member
MHB Math Helper
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,...,vn} be a basis for V and b={w1,w2,....,wm} 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.

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Fernando Revilla

Well-known member
MHB Math Helper
$(a)\;$ 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 $\dim (\ker T)=n-\mbox{rank }[T]_{BA}$.