Additional Vector Space Problems
1. Suppose that V is a vector space and W is a vector subspace of it.
Let WP denote the set of all vectors which are orthogonal to
vectors in W (their dot product is 0). Prove that WP is also
a vector space.
2. Consider all solutions to the general second order differential equation
a(x)y’’ + b(x) y’ + c(x)y = 0
where ‘ denotes derivative wrt x
Prove this is a vector space.
3. Consider all nxn matrices with non zero determinant. Is this set
a vector space?? prove or disprove
4. Is the intersection of two vector spaces a vector space?? prove or
disprove.
5. Is the set of all nxn symmetric matrices a vector space?? prove or
disprove.
6. Consider an nxm matrix A and the set of all linear combinations of
columns of A (call it V). Prove V is a vector space. Then prove
that the problem Ax = b has a solution if and only if b is in V
7. Define a function to be “even” if it satisfies f(-x) = f(x) and odd
if f(-x) = - f(x).
Prove that the set of all even functions is a vector space. Prove that
the set of all odd functions is a vector space.
Prove that ANY function can be written as the sum of an even + an
odd function
8. Define an nxn U matrix to be “orthogonal” if it satisfies UtU = I
in other words, its transpose is its inverse.
Is the set of all nxn orthogonal matrices a vector space???