Linear Algebra - Finding a Basis

[CornMuffin]

Homework Statement

I am having trouble finding a basis in a given vector space.

I understand how to find a basis of Rn, just find linearly independent vectors that span Rn

But how would i find a basis of the set of 3x3 symmetric real matrices?
Or Find a basis of real polynomials of degree less than or equal to 3?

If i understand more about a basis, then I might be able to do this.

Homework Equations

A set of vectors B in a vector space S is a basis of S iff B is a linearly independent spanning set of S.

The Attempt at a Solution

 

[CompuChip]

A set of vectors B in a vector space S is a basis of S iff B is a linearly independent spanning set of S.

... which is just fancy talk for (basically): if you take any element of S, you can express it as a linear combination of the elements of B.

For example, a basis for the space of all 2 x 2 matrices would be
$$\left\{ B_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, B_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, B_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, B_4 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\}$$
because you can write any matrix
$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$
as a linear combination, namely:
$$A = a B_1 + b B_2 + c B_3 + d B_4$$

 

[CornMuffin]

thank you, that helps