# Constructing isomorphisms

## Homework Statement

Construct an isomorphism from a 2 by 2 symmetric matrix to R^3.

N/A

## The Attempt at a Solution

I know that for a transformation to be an isomorphic, it must be one-to-one and onto. Would the transform be T:A->R^3 and I would have to choose a general matrix A to test?

How would I test it not knowing how the transform is mapped?

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An isomorphism between which structure?? Vector spaces??

Anyway, given a symmetric matrix

$$\left(\begin{array}{cc} a & b\\ b & c \end{array}\right)$$

what element of $\mathbb{R}^3$ would you associate with this matrix??

An isomorphism between which structure?? Vector spaces??

Anyway, given a symmetric matrix

$$\left(\begin{array}{cc} a & b\\ b & c \end{array}\right)$$

what element of $\mathbb{R}^3$ would you associate with this matrix??
Yes vector spaces. What do you mean by what element? Would R^3 simply be some vector v=(v1, v2, v3)?

Yes vector spaces. What do you mean by what element? Would R^3 simply be some vector v=(v1, v2, v3)?

Yes, elements of $\mathbb{R}^3$ would just be vectors (a,b,c).

So then I would just check the the nullspace and the dimension of the range? What would be the form of my answer? A matrix?

You still need a suggestion for what your isomorphism actually does. To which matrix would you map (a,b,c)?? That is: if I give you three real numbers, how would you make a symmetric matrix out of it??