# Isomorphism: subspace to subspace?

## Homework Statement

We're looking at a mapping from P2 (polynomials of degree two or less) to M2(R) (the set of 2x2 real matrices). The nuance here is that the transformation into the matricies is such that its basis consists of only three independent matrices, making its dimension 3. This means that our transformation maps from P2 (dim = 3) to M2(r) (dim = 3 in this case)
Can a mapping to a subspace make the transformation an isomorphism?

## The Attempt at a Solution

micromass
Staff Emeritus
Homework Helper
Yes, this mapping might be an isomorphism on the sibspace. Do you know whether it is surjective?? This is already enough for showing it an isomorphism.

Indeed: if V and W both have equal dimension then a map $T:V\rightarrow W$ is an isomorphism if and only if it is surjective and linear (or injective and linear). This is the alternative theorem.

The transformation itself is
P2 →M2(ℝ)
T(ax2 + bx + c) → Matrix(a11 = -b-a, a12 = 0, a21 = 3c-a, a22 = -2b)

Last edited:
micromass
Staff Emeritus
Homework Helper
Yes, so is the range three dimensional??

Yes, the range itself is a three dimensional subspace of the four dimension space of 2x2 real matrices. I'm just not sure if it's an isomorphism.

micromass
Staff Emeritus