Midterm 1 - MATH 221 (2017T2)

Problem 1

Solve the following linear system and write the solution in parametric vector form.

Problem 2

Write down all equations, if any, satisfied by the components , , and of that is in the span of , , and . Is the span the entire ? Why or why not?

All the solutions satisfy the equation . It does not span as there are points in which do not satisfy that equation for example .

Problem 3

Consider , , and , for some constant .

(a). Find All Values of such that the Vectors , , and Are Linearly Dependent

(b). Choose a Value of such that , and Are Linearly Dependent, and Write down a Linear Dependence relation

Problem 4

Suppose is a linear transformation such that , and

(a). What is

(b). What is (Hint Write it as a Linear combination)

Problem 5

Check which of the following statements are true

[ ] A. matrix has seven rows.
[x] B. The matrix is in reduced echelon form.
[ ] C. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
[ ] D. A linear system is unique if and only if it has exactly one solution.
[ ] E. The equation is consistent if the augmented matrix has a pivot position in every row.
[x] F. A homogeneous linear system is always consistent.
[x] G. The columns of any matrix are linearly dependent.
[ ] H. Let be an matrix and be a nonzero vector. If the solution set of is infinite, then the solution set of is also infinite.
[x] I. The transform defined by .
[ ] J. If is an matrix, then the range of the transformation is .