# Linear transformations

1. Dec 17, 2009

### phrygian

1. The problem statement, all variables and given/known data

(124) If a linear transformation T : R3 -> R5 is one-to-one, then
(a) Its rank is five and its nullity is two.
(b) Its rank and nullity can be any pair of non-negative numbers that add
up to five.
(c) Its rank is three and its nullity is two.
(d) Its rank is two and its nullity is three.
(e) Its rank is three and its nullity is zero.
(f) Its rank and nullity can be any pair of non-negative numbers that add
up to three.
(g) The situation is impossible.

(125) If a linear transformation T : R3 -> R5 is onto, then
(a) Its rank is five and its nullity is two.
(b) Its rank is two and its nullity is three.
(c) Its rank is three and its nullity is zero.
(d) Its rank and nullity can be any pair of non-negative numbers that add
up to three.
(e) Its rank is three and its nullity is two.
(f) Its rank and nullity can be any pair of non-negative numbers that add
up to five.
(g) The situation is impossible.

2. Relevant equations

3. The attempt at a solution

These two problems on my practice test have me completely stumped, could someone help shed some light?

I understand the definitions of onto and one-to-one, but don't understnad how to connect this to the null space of a linear transformation from Rm to Rn?

2. Dec 17, 2009

### HallsofIvy

Staff Emeritus
Saying that a linear transformation is "one-to-one" means that only one "u" is mapped to a specific "v", right? In particular, that means only one vector is mapped to the 0 vector. Since any linear transformation maps the 0 vector to the 0 vector, a one to one mapping maps only the 0 vector to the 0 vector.

That is, a linear transformation is one to one if and only if its nullspace is the "trivial" space consisting only of the 0 vector.