Math 322

1st Midterm

Spring 2001

Dr. Wilson

 

1. Solve the following system of two equations in two unknowns by finding the inverse of the coefficient matrix and multiplying it by the column of answers.

2x + 3y = 6

3x + 4y = 7

2. Find the dimension and a basis for the row space of the following matrix by row reducing it. Verify that your alleged basis is in fact a basis.

3. Let S be a set and let F be a field. We know that V, the set of all functions from S to F, is a vector space with the operations

(f + g)(x) = f(x) + g(x)

(af)(x) = a.f(x)

where x is an element of S, f and g are functions from S to F, and a is a scalar in F.Let s be a fixed element of S. Prove that

W = {f: S -> F | f(s) = 0 }

is a subspace of V.

4. Let V and W be vector spaces over a field F. A function

T: V -> W

is called a linear transformation if

Prove that the set of linear transformations forms a vector space over F.