Ok here's the deal. I'm returning to Math after a break and I need all the help I can get. I'm not looking for answers just hints/pointers to help me address my HW and advance my understanding of Linear Algebra. About me...former B.S. Biochem major making the transition to ChemE. The book is the 10th Edition Elementary Linear Algebra by Anton.

I will be answering the prob and stating why I think it so or why I have no freaking idea

4.2 probs 4, 8 ac, 10 ac, 14, and T/F(k)

4) Which of the following are subspaces of F(-infinity, infinity)

a) all fxns f in F(-infin, infin) for which f(0)= 0

yes because that satisfies the zero vector

b) all fxns f in F (-infin, infin) for which f(0) = 1

no b/c doesn't satisfy zero vector

c) all fxns f in F(-infin, infin) for which f(-x) = f(x)

yes b/c needs to satisfy for negative scalar

d) all polynomials of degree 2

???????

8) express the following as linear combos ofu= (2,1,4),v= (1,-1,3) andw= (3,2,5)

a) (-9, -7, -15)

9 <2,1,4> + -7 <1,-1,3> + -15 <3,2,5> = <-34, -14, -92>

c) (0,0,0)

0 < > + 0 < > + 0 < > =0

10) express the vector as a linear combo of p_{1}= 2 + x + 4x^{2}, p_{2}= 1 - x + 3x^{2 }, p_{3}= 3 + 2x + 5x^{2}

a) -9 - 7x -15x^{2 }-9 - 7x -15x^{2}= (2a1 + a2 + 3a3) + x(a1 - a2 + 2a3) + x^{2 }(4a1 + 3a2 + 5a3)

2 1 3 -9

1 -1 2 -7

4 3 5 -15

c) 0

0 = (2a1 + a2 + 3a3) + x(a1 - a2 + 2a3) + x^{2 }(4a1 + 3a2 + 5a3)

2 1 3 0

1 -1 2 0

4 3 5 0

14) letf= cos^{2}x andg= sin^{2}x. Which of the following lie in the space spanned byfandg?

a) cos2x b) 3 + x^{2}c) 1 d) sin x e) 0

I tried (f,g) = k_{1}f+ k_{2}g= cos^{2}x, sin^{2}x = k1(cos^{2}x) + k2(sin^{2}x) .............. : (

I don't have the slightest. I know that I'm trying to prove that a-e are subspaces enclosed by f and g and must have zero vector but I'm at a loss. I thought maybe cos2x + sin2x = 1 might help but I got nothing

True or False (K) and explain why

The polynomials (x-1), (x-1)^{2}and (x-1)^{3 }span P_{3}?

I started out p_{1}= x -1, p_{2}= (x - 1)^{2}and p_{3}= (x - 1)^{3}

Please Help