## 1st Midterm

## Spring 2001

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.