How to do Linear Algebra 1 The set B 1 t2 t t 2 t t2 is a b

How to do Linear Algebra ??

1. The set B 1 t2, t t 2 t t2) is a basis for P2. Find the coordinate vector of p(t) 1 3t 6t2 relative to B. That is, find [p t)]

Solution

(1) To find [p(t)]_B , we have to express p(t) as a linear combination of the basis vectors.

So, Let a, b, c such that-----

1+3t-6t² = a(1-t²)+b(t-t²)+c(2-t+t²)

1+3t-6t² = (a+2c)+(b-c)t+(-a-b+c)t²

So by comparison----

-a-b+c = -6_____(1)

b-c = 3_____(2)

a+2c = 1____(3)

Add (1) and (3)

-b+3c = -5______(4)

Add (2) and (4)----

2c = -2 so c = -1

From (3) a = 3

From (1). b = 2

So, [p(t)]_B = [3, 2, -1]

(2) Let (2-3t) = p(5t+t²)+q(1-8t-2t²)+r(-3+4t+2t²)

2-3t = (q-3r)+(5p-8q+4r)t+(p-2q+2r)t²

So, by comparison ---

q-3r = 2

5p-8q+4r = -3

p-2q+2r = 0

So, solution not possible.

Hence all vectors are linearly independent and span P₂.