Additional Vector Space Problems



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???

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