Linear transformations flashcards

Question 1

What is a linear transformation $T:V	o W$?

Accepted Answer

A function $T$ such that for all $u,v\in V$ and scalars $c$, $T(u+v)=T(u)+T(v)$ and $T(cu)=cT(u)$.

Question 2

What is the image (range) of a linear transformation $T:V	o W$?

Accepted Answer

$\operatorname{Im}(T)=\{T(v): v\in V\}$; it is a subspace of $W$.

Question 3

State the rank-nullity theorem for $T:V	o W$ with $V$ finite-dimensional.

Accepted Answer

$$\dim(V)=\dim(\ker T)+\dim(\operatorname{Im} T).$$ Here $\dim(\operatorname{Im} T)$ is the rank and $\dim(\ker T)$ is the nullity.

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