Question

Consider the polynomials p1(t) = 1 + t , p2(t) = 1 -t , and p3(t) = 2 (for all t). By inspection, write a linear dependence relation among p1, p2, and p3. Then find a basis for Span{ p1 , p2 , p3 }.

Answer 1

Answer:

$\{p_1,p_2\} is a basis of Span\{p_1,p_2,p_3\}$

Step-by-step explanation:

Given the polynomials:

$p_1(t) = 1 + t\\ p_2(t) = 1 -t\\p_3(t) = 2$

On Inspection

$(1+t)+(1-t)=1+1+t-t=2\\ Therefore:\\p_1(t)+p_2(t)=p_3(t)$

By the Spanning Theorem  

If one vector in S is a linear combination of the others, we can delete it and get a subset (one vector smaller) $S' \subseteq S$ that has the same span.

Therefore, since $p_3(t)=p_1(t)+p_2(t)$

$Span\{p_1,p_2,p_3\}=Span\{p_1,p_2\}$

$p_1 and p_2$ are linearly independent because $p_1$ cannot be written in terms of $p_2$.

Therefore, $\{p_1,p_2\} is a basis of Span\{p_1,p_2,p_3\} as required.$