Finding a Basis for V: Proving Linear Independence and Determining a Basis for V

[misterau]

Homework Statement

Let V be the vector space spanned by v1 = cos^2(x) , v2 = sin^2(x) , v3 = cos(2x) .
Show that
{v1 ,v2 ,v3} is not a basis for V , then find a basis for V .

Homework Equations

The Attempt at a Solution

(-1)*cos^2(x) + (1)*sin^2(x) + (1)*cos(2x)=0
{v1 ,v2 ,v3} is not linearly independent, so is not a basis for V.

I am not sure how to do the next part of the question, "find a basis for V" .
I am thinking its probably {v1,v2}. As v1 and v2 are linearly independent. However how do I show this set spans V?

 

[Mark44]

V is the space spanned by v1 = cos^2(x), v2 = sin^2(x), and v3 = cos(2x). Since cos(2x) is a linear combination of v1 and v2, we can remove v3 without removing any of the vectors in V.

What does it mean (i.e., the definition) to say a space is spanned by a set of vectors?

 

[misterau]

V is the space spanned by v1 = cos^2(x), v2 = sin^2(x), and v3 = cos(2x). Since cos(2x) is a linear combination of v1 and v2, we can remove v3 without removing any of the vectors in V.

What does it mean (i.e., the definition) to say a space is spanned by a set of vectors?

To span a space means that every vector in the space can be written as a linear combination in the set.

 

[ksinclair13]

Write v (an arbitrary vector in V) as a linear combination of v1, v2, and v3 and then see if you can write v as a linear combination of just v1 and v2. Hint: v3 = v1-v2.