If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. W is the set of all vectors of the form [a - 4b 5 4a + b -a - b], where a and bare arbitrary real numbers. a. [1 5 4 -1], [-4 0 1 -1]
b. [1 0 4 -1], [-4 5 1 -1]
c. [1 0 4 -1], [-4 0 1 -1], [0 5 0 0]
d. Not a vector space

Choice d. The set of vectors: isn't a vector space over .

Explanation:

Let a set of vectors to be a vector field over some field (for this question, that "field" is the set of all real number.) The following must be true:

Note that in the general form of a vector in , the second component is a always non-zero. Because of that non-zero component,

Assume by contradiction that is indeed a vector field. Therefore, it should contain a zero vector. Let denote that zero vector. For all , .

Using the definition of set : , there exist real numbers and , such that:

.

Hence, is equivalent to:

.

Apply the third property that is closed under scalar multiplication. is indeed a real number. Therefore, if is in

Therefore:

.

Apply the second property and add to both sides of . The left-hand side becomes:

.

The right-hand side becomes:

.

Therefore:

.

However, isn't a member of the set . That's a contradiction, because was supposed to be part of .

Hence, isn't a vector space by contradiction.