Page 165 - A Course in Linear Algebra with Applications

5.3: Operations with Subspaces 149

2. Let U and W denote the sets of all n x n real symmetric
and skew-symmetric matrices respectively. Show that these
are subspaces of M n (R), and that M n (R) is the direct sum of
U and W. Find dim(U) and dim(W).
3. Let U and W be subspaces of a vector space V and suppose
that each vector v in V has a unique expression of the form
v = u + w where u belongs to U and w to W. Prove that
V = U e W.
4. Let U, V, W be subspaces of some vector space and suppose
that U C W. Prove that

(u + v) n w= u + (v n w).

5. Prove or disprove the following statement: if U, V, W are
subspaces of a vector space, then (U + V) n W = (U D W) +
(VHW).

6. Suppose that U and W are subspaces of Pi4(R) with
dim(C/) = 7 and dim(W) = 11. Show that dim(U n l f ) > 4 .
Give an example to show that this minimum dimension can
occur.
7. Let M be the matrix

/ 3 3 2 8\
1 1 - 1 1
1 1 3 5
\ - 2 4 6 8 /

and let U and W be the subspaces of R 4 generated by rows 1
and 2 of M, and by rows 3 and 4 of M respectively. Find the
dimensions of U + W and U fl W.
8. Define polynomials

3 2 3
/i = 1 - 2x + x , f 2 = x + x - x .