Question

Find an orthonormal basis of the subspace S = {x E R^4: x1 + x2 +x3 +x4 = x1 - x2 -2*x3 + x4 = x2 + 3*x3 =0}. What is the dimension on S?

Answer 1

Well a basis for the subspace
S1 = { x in R^4 | x1 + x2 +x3 + x4 = 0}
is
B1 = { (1,0,0,-1), (0,1,0,-1), (0,0,1,-1) }

In other words S1 = span(B1)

Now find expressions for S2, B2, S3, B3.
And then observe that by definition
S = S1 n S2 n S3

And see if you can see the basis. In any case, you should be able to see the dimension of S very quickly.

Formalizing all of this, row reduce homogeneous equations which give S.

Answer 2

Hi James, thank you very much for taking the time to answer my question, I'm sort of new this idea and wondering if you could explain what S1 n S2 n S3 means and possibly what your last sentence means? I'm sorry again, I'm new to all this.

Answer 3

S1 n S2 n S3 means the intersection of these three subspaces

Answer 4

Also, you have three equations
x1 + x2 + x3 +x4 = 0
x1 - x2 -2*x3 + x4 = 0
x2 + 3*x3 =0

Solve them as best you can and the number of degrees of freedom you have will tell you the dimension of S and you will also see very quickly a basis

Answer 5

Damn, James is fast :P

Answer 6

Haha, I appreciate it anyway Cptn. James, thank you very much! I can see why you've been bestowed title of Superhero.

Answer 7

If you spend enough time on this site you'll learn that mathematical ability and level have roughly zero correlation. But thanks.

Answer 8

True, true

Answer 9

Haha, so I assume I should put the final answer into row reduced echelon form and then the basis and dimensions will shake out of that?

Answer 10

No ... the linear algebra part of row reduction is a computation you need to do to get to the answer. Do the row reduction and then think about what it means.

Answer 11

Remember that the rank of the matrix is the order of the basis of the generated space

Answer 12

So what I'm thinking now is the after I find the rref of the matrix, the resulting nonzero columns will be the basis

Answer 13

and because I have three equations, the rank of the matrix will be three or less than three when I'm done

Answer 14

Sorry, what does rref mean? (english is not my native language)

Answer 15

Oh I'm sorry, row reduced echelon form

Answer 16

Oh, thank you :P I should have seen that

Answer 17

But I believe that's the right way to go about this? Find the row reduced form of the matrix, the resulting nonzero terms are the basis and the rank should be three or less than three? The rank part relationship and how to find it I'm silly fuzzy on.