Making a Set a Basis: Is V Spanned?

  • Thread starter torquerotates
  • Start date
  • Tags
    Basis Set
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.
  • #1
torquerotates
207
0
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?
 
Physics news on Phys.org
  • #2
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 [itex]\cos^2x = 1 - \sin^2x[/itex].
 
  • #3
torquerotates said:
does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

try evaluating at x=0 and x=[itex]\pi/2[/itex]
 

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.

Similar threads

I S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?
    • Linear and Abstract Algebra
Replies
5
Views
871
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
    • Linear and Abstract Algebra
Replies
3
Views
2K
I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
861
I Non orthogonal basis and the lines of its coordinate grid
    • Linear and Abstract Algebra
Replies
9
Views
186
I Standard basis and other bases...
    • Linear and Abstract Algebra
Replies
9
Views
562
MHB Basis for set of solutions for linear equation
    • Linear and Abstract Algebra
Replies
6
Views
1K
MHB Diagonalizable transformation - Existence of basis
    • Linear and Abstract Algebra
2
Replies
52
Views
2K
I Orthonormal basis expression for ordinary contraction of a tensor
    • Linear and Abstract Algebra
Replies
2
Views
903
MHB Set of 2-dimensional orthogonal matrices equal to an union of sets
    • Linear and Abstract Algebra
Replies
9
Views
867

Hot Threads

Recent Insights

Back
Top