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

Answer 2

A linear dependence relation among the polynomials p₁, p₂, and p₃ is given as p₁ + p₂ = p₃

What is polynomial?

Polynomial is an algebraic expression that consists of variables and coefficients. Variable are called unknown. We can apply arithmetic operations such as addition, subtraction, etc. But not divisible by variable.

Consider the polynomials are

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

a linear dependence relation among p₁, p₂, and p₃ will be

Let add the  p₁ and p₂, we have

1 + t + 1 -t = 2

We get the polynomial p₃.

Thus the linear relation is p₁ + p₂ = p₃

More about the polynomial link is given below.

https://brainly.com/question/17822016