# Basis question

This is the problem that im working on.

Find a basis for the vector space of all 3x3 symetric matricies.

Is this a good place to start

111
110
100

using that upper triangular

then spliting it into the set.

100 010 001 000 000 000
000 000 000 100 010 000
000 000 000 000 000 100

would that work?

i know that it has to be linearly independent and span.

Last edited:

matt grime
Homework Helper
Are any of the matrices you wrote out symmetric?

The matrix you started from isn't upper triangular, by the way. Why would you want to start with that matrix anyway? A symmetric matrix is one satisfying A=A^t, so take the 3x3 matrix

[a,b,c]
[d,e,f]
[g,h,j]

and transpose it, equate the two, what conditions do you now have on a,b,..,j?

I choose

111
110
100

because it was equal to its transpose.
I know that b is equal to to d and so on for the transpose you mention.

I choose the six randomly as a guess.

100
000
000 becuase a doesnt move

010
100
000 those have to be there

001
000
100 same here

000
000
001 and this

000
001
010

I guess that im just looking for a pattern. Those seem better, I feel like there should be six though.

Homework Helper
One matrix is missing.

000
010
000

right?

Homework Helper
Yes, right.

Ok. So here is where my new issue arrises.

For this set of matricies to be a BASIS for all symertric 3x3 matricies then it has to linearly independent... right?

So I took my set.

100 010 000 001 000 000
000 100 010 000 001 000
000 000 000 100 010 001

and set it equal to a 3x3 zero matrice, and set up a system with scalars.
They end up all equaling zero... the trivial solution. Nice.

But they have to check out for span also to be considered a basis right?

That means that each one has to be a linear combination of the other five remaining right?

Well for some of them I get 0 = 1 for a solution

Anyone know where im going wrong?

Thanks, Rob

matt grime
Homework Helper
Yes. It is this line:

But they have to check out for span also to be considered a basis right?

That means that each one has to be a linear combination of the other five remaining right?

Reread the definition of span. Cos you've just written something that implies linear dependence. It spans if every symmetric matrix can be written as a combination of them, and that is obviously true.

This is the problem that im working on.

Find a basis for the vector space of all 3x3 symetric matricies.

would any one help me solve this problem?

I read all posts above but still confuse.

Much appreciate.

matt grime
Homework Helper
What is consfusing you? What is the generic symmetric matrix? Therefore what is a basis?

I know 3x3 symmetric matrix formed by
a b c
d e f
g h i
and b=d c=g f=h
but I don't know how to start on this case.
What I'd write at beginning on my homework paper?

matt grime
Homework Helper
Write that as a sum of matrices with lots of zeroes.

That array just refuses to cooperate...

daniel_i_l
Gold Member
To find a basis you just do three steps:
1) find the generic matrix (or vector in the general case)
2) break it up into a number of matricies where each one containes only one type of parameter with everything else zeros(but no two contain the same type)
3) turn the parameters in each matrix into ones.