Consider the polynomials Bold p 1 (t )p1(t) equals = 3 3 plus + 8 8t, Bold p 2 (t )p2(t) equals = 3 3 minus − 8 8t, and Bold p 3 (t )p3(t) equals = 6 6 (for all t). By inspection, write a linear dependence relation among Bold p 1 p1, Bold p 2 p2, and Bold p 3 p3. Then find a basis for Span StartSet Bold p 1 comma Bold p 2 comma Bold p 3 EndSet Spanp1, p2, p3.

Apparently .

One possible basis for this span is .

Step-by-step explanation:

A set of vectors is linearly-dependent if one of the vectors is a linear combination of the others.

Alternatively, to show linear dependence, show that the equation has a non-trivial solution (where at least one of the three scalars , , and is non-zero.)

For this set of polynomials, it can be shown that:

, or equivalently,

.

Either way, the set would be linearly-dependent.

implies that is a linear combination of and . Therefore, is in the span of and (in other words, .) Hence:

.

Therefore, any basis of the set would also be a basis of the set .

On the other hand, it can be shown that and are linearly-independent. Therefore, should have a dimension of . As a result, there should be exactly linearly-independent vectors in a basis of .

There are many different choices for the basis of . One possible choice is the set , which is equal to . Make sure that this set is indeed linearly-independent and contains two vectors.

Because it has already been shown that , should be a basis of the set , as well.

ok then what happened.