Linear algebra homework problem Let f cos2x and g sin2x Wh

Linear algebra homework problem: Let f = cos²⁽x) and g = sin²(x). Which of the following lie in the space spanned by f and g? Show why or why not? Please show work and provide full description

a) cos(2x)

b) 7

c) x²

^{d) 0}

^{e ) sin(x)}

Edit: So I know that there are three properties that need satisfies, if there is subspace.

1) The zero vector of H is in H.

2) H is closed under vector addition. That is, for each u and v in H, the sum u+v is in H.

3) H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H.

Solution

To check if any of them lie in the space spanned by f and g we need to check whether they can be written as a linear combination of f and g.

a) As cos2x = cos²x - sin²x = 1.f + (-1).g ; hence cos2x is spanned by f and g.

b) x²

any function that lie in space spanned by f & g is periodic since they are periodic but  x² isn\'t. therefore it is not in space spanned by f & g.

an other method to prove it is:-

if x² lie in space spanned by f & g then,

x² = a. cos²x +b. sin²x

take x=0, then 0= a + 0 => a= 0

i.e. x² = b. sin²x

take x= pi/2 => (pi/2)² = b

take x = pi/4 => (pi/4)² = a/2 + b/2

=>   pi² / 16 = 0 + pi² / 8 ...[ substituting values of a & b]

Contradiction. Hence x² is not in the span of f& g.

d) 0 . yes, taking a & b = 0 in the above expression.

e) sinx . No

there are many ways to see this. sinx has smallest period 2, while cos²x and sin²x have smallest period .

you can also solve it using above method taking

sinx = a.cos²x +b. sin²x and try to find contradiction as we did above.

PS: what you wrote about the subspace in the end is true but it is not relevant here.