Problem 1 Consider the vector space P4 of polynomials of deg

Problem 1. Consider the vector space P4 of polynomials of degree at most 4. (1A) What is a basis for P4? (1B) What is the dimension of P4?

Solution

1A

Basis for P4 is {1,t,t^2,t^3}

1B

Basis has 4 vectors so dimension is 4

1C

Clearly span(B)=W

We need to prove linear independence of W

Let, a,b so that

a(t^4+t^3+t)+b(2t^4-t^3-2t)=0

a+2b=0

a-b=0

a-2b=0

HEnce, a=b=0

So, B is a linearly independent set

So, B forms basis for W