# Making a Set a Basis: Is V Spanned?

• torquerotates
In summary, the set S={cosx^2, sinx^2, cos(2x)} spans V, but is not a basis because the third vector is a linear combination of the first two. The first two vectors are not linearly independent as cos^2x = 1 - sin^2x. However, evaluating at x=0 and x=\pi/2 can help determine if they span V.
torquerotates
Ok, so I have the set S={cosx^2, sinx^2, cos(2x)} that spans V.

So obviously this is not a basis. Because the third is a linear combination of the first; ie. cosx^2-sinx^2=cos(2x). But if I were to take the first 2 vectors, would they span V? That is does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

torquerotates said:
Ok, so I have the set S={cosx^2, sinx^2, cos(2x)} that spans V.

So obviously this is not a basis. Because the third is a linear combination of the first; ie. cosx^2-sinx^2=cos(2x). But if I were to take the first 2 vectors, would they span V? That is does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

Do you mean: Are the first two vectors linearly independent?

Use $\cos^2x = 1 - \sin^2x$.

torquerotates said:
does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

try evaluating at x=0 and x=$\pi/2$

## 1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that every vector in the vector space can be written as a unique combination of the basis vectors.

## 2. How do you know if a set is a basis?

A set is a basis if the vectors in the set are linearly independent and span the vector space. This means that none of the vectors in the set can be written as a linear combination of the other vectors, and every vector in the vector space can be written as a unique combination of the basis vectors.

## 3. Can a set with fewer vectors than the dimension of the vector space be a basis?

No, a set with fewer vectors than the dimension of the vector space cannot be a basis. A basis must have the same number of vectors as the dimension of the vector space in order to span the entire space.

## 4. Is every vector space spanned by a basis?

Yes, every vector space can be spanned by a basis. This is one of the fundamental properties of vector spaces and is known as the basis theorem.

## 5. Can a set with linearly dependent vectors be a basis?

No, a set with linearly dependent vectors cannot be a basis. A basis must consist of linearly independent vectors in order to uniquely span the vector space.

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