## 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 ve

Question

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

in progress 0
5 months 2021-08-16T05:03:34+00:00 1 Answers 25 views 0

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:

• The set of vectors includes the identity element . In other words, there exists a vector such that for all , .
• should be closed under vector addition. In other words, for all , .
• should also be closed under scalar multiplication. In other words, for all and all “scalar” (in this question, the “field” is the set of all real numbers, so can be any real number,) .

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.