Problem 1

Determine if the columns of the following matrices form a linearly independent set. Justify each answer.

(a).

Yes they form a linearly independent set as there is some combination of them which equals the zero vector.

Problem 2

Find the values of for which the vectors are linearly dependent. Justify each answer.

(a). , , and

Problem 3

Mark each statement True or False.

• [x] Two vectors are linearly dependent if and only if they lie on a line through the origin.
• [ ] If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
• [x] If and are linearly independent, and if is in , then is linearly dependent.
• [ ] If a set in is linearly dependent, then the set contains more vectors than there are entries in each vector.

Problem 4

Describe the possible echelon forms of the matrix, with representing pivot positions and representing any number (zero or nonzero).

(a). A is a Matrix, , such that is Linearly Independent and is not in

Problem 5

Prove that the following statement is either true or false.

If are in and is not a linear combination of , , , then is linearly independent.

False consider the vectors