# Thread: Algebra - Vector Spaces

1. ## Algebra - Vector Spaces

In the following, Mn,n(R) denotes the vector space of real
n × n matrices and, for A Ɛ Mn,n(R), A^T denotes the transpose of A.

1. For each of the following, either use the subspace test to show that the given
subset,
W, is a subspace of V or explain why the given subset is not a subspace
of
V .

(a)
V = M2,2(R) and W = {A Ɛ V | A^T = A}.
(b) V = R^3 and W = {(2t, 3t,5t) | t ƐR}.

Any ideas on how to start? I'm useless with Vector spaces and am trying to learn it now, but i really don't know how to solve the above. Please be of guidance and assisstance.

2. Dear James1220,

you must prove if you take two element from W (mark: w1 and v2) than their linear comination w belong to W also. Linear combination means p*w1 + q*w2 where p and q arbitary real (or scalar). The operate * and + means same as in the original vector space.

For exapmle. Let V = R^3 is a vector space. We can express an element as a triplet (x, y, z).
Prove that elemnts with condition z=0 is subspace.
Proof: Let w1 and w2 element of W. So we can write
w1 = (x1, y1, 0) and w2 = (x2, y2, 0).

Image the linear combination:
$\displaystyle w = p*w1 + q*w2 = p*(x1, y1, 0) + q*(x2, y2, 0) = (p*x1+q*x2, p*y1+q*xy, 0)$
Is w element of W? Yes, of course since the 3th koordinate is null again.

3. Thank you. So i'm guessing the fact that they are null, it means V is linearly independent? Is this the solution to part (b)?

How would i do question (a), which has information on the transpose of A?

4. Thank you. So i'm guessing the fact that they are null, it means W is a subspace of V??? Is this the solution to part (b)?

How would i do question (a), which has information on the transpose of A?

5. (A + B)^T = A^T + B^T (check out on a concrate matrix)

You said: "they are null"
What is null? You must take two element: w1 = (2t, 3t, -5t) and w2 = (2s, 3s, -5s) and go on...