# A vector space W over the real numbers is the set of all 2 x 2

ZWAN

## Homework Statement

A vector space W over the real numbers is the set of all 2 x 2 Hermitian matrices. Show that the map T defined as:

T(x,y,z,t) =
[t+x y+iz]
[y-iz t-x]

from R4 to W is an isomorphism.

## The Attempt at a Solution

I know that for the map to be isomorphic it has to have the following properties:
- one to one (injective) and onto (surjective)
- ker(T) = {0} and range(T) = W
- The inverse map T^-1 has to exist
- dimension of both R4 and W has to be the same.

I don't really know how to apply these properties in showing that the map is isomorphic. Any help would be appreciated. Thanks in advanced!

## The Attempt at a Solution

Staff Emeritus
Gold Member

An isomorphism is by definition a linear bijection, and a bijection is by definition a map that's both injective and surjective. So there are three things you must verify:

1. T is linear.
T(ax+by)=...(use what you know about T here)...aTx+bTy.​
2. T is injective.
Show that Tx=Ty implies x=y.​
3. T is surjective.
Let z be an arbitrary 2×2 hermitian matrix. Prove that there's an x in ℝ4 such that Tx=z.​

Gold Member
MHB

a hint, sort of:

can you find linearly independent (over R, not C) 2x2 matrices in Mat2x2(C), A,B,C,D such that:

T(x,y,z,t) = xA + yB + zC + tD?

linearity is easy to prove, if so, and so is bijectivity.

(why? because if {A,B,C,D} is LI, it's a basis of something, right?)

another way to prove injectivity:

show ker(T) = {(0,0,0,0)}.

a pre-image of w in W should not be hard to find. figure out what y and z have to be, first, and then tackle x and t. you should get a system of two equations in x and t, easy to solve.