Orthonormal Basis Homework: True/False

In summary, the set of vectors B={(-1,-1,1,1),(1,0,0,0),(0,1,0,0),(-1,-1,1,-1)} is not an orthonormal basis for Euclidean 4-space \mathbb{R}^4 because at least one vector is not a unit vector and some vectors are not orthogonal. The use of the word "Euclidean" implies the use of the usual inner product.
  • #1
DanielFaraday
87
0

Homework Statement



True/False:

The set of vectors [tex]B={(-1,-1,1,1),(1,0,0,0),(0,1,0,0),(-1,-1,1,-1)}[/tex] is an orthonormal basis for Euclidean 4-space [tex]\mathbb{R}^4[/tex].

Homework Equations


None

The Attempt at a Solution



I said false because [tex]\langle (-1,-1,1,1),(-1,-1,1,1) \rangle =2\ne1[/tex], which shows that at least one vector in this set is not a unit vector.

However, I'm not sure if I'm supposed to use the usual definition for the inner product. Is this implied by the word "Euclidean"?
 
Physics news on Phys.org
  • #2
That looks right. Some of the vectors aren't orthogonal either. "Euclidean" would imply the usual inner product. But even if they left the word "Euclidean" off, I would still use the usual inner product, just because they didn't tell you to use a different one.
 
  • #3
Perfect. Thanks!
 

1. What is an orthonormal basis?

An orthonormal basis is a set of vectors in a vector space that are both orthogonal (perpendicular) to each other and have a unit length of 1.

2. What is the importance of orthonormal basis in linear algebra?

Orthonormal basis is important in linear algebra because it allows for vector spaces to be easily represented and manipulated using simple operations such as dot products and projections. It also simplifies calculations and makes it easier to solve systems of linear equations.

3. How do you determine if a set of vectors form an orthonormal basis?

To determine if a set of vectors form an orthonormal basis, you must check that each vector is orthogonal to every other vector in the set and that each vector has a unit length of 1. You can also check if the set of vectors is linearly independent, which is a necessary condition for forming an orthonormal basis.

4. Can a set of vectors have more than one orthonormal basis?

Yes, a set of vectors can have more than one orthonormal basis. This is because there can be multiple sets of vectors that are both orthogonal and have a unit length of 1. The basis vectors themselves may be different, but they still represent the same vector space.

5. True or False: A matrix with orthonormal columns is always an orthonormal basis.

False. A matrix with orthonormal columns does not necessarily form an orthonormal basis. It is possible for the columns to be orthogonal to each other, but not have a unit length of 1. An orthonormal basis must have both orthogonal vectors and a unit length of 1.

Similar threads

I cannot make the Gram-Schmidt procedure work
    • Calculus and Beyond Homework Help
Replies
16
Views
1K
Dimension of orthogonal subspaces sum
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Minimize integral using orthogonal basis
    • Calculus and Beyond Homework Help
Replies
2
Views
344
I Orthonormal basis expression for ordinary contraction of a tensor
    • Linear and Abstract Algebra
Replies
2
Views
895
Finding bases for ##W_1\cap W_2## and ##W_1+W_2##
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
    • Linear and Abstract Algebra
Replies
3
Views
2K
Linear Combination Proof of Orthonormal basis
    • Calculus and Beyond Homework Help
Replies
11
Views
2K
Orthogonality and Orthonormality: Take 2
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
I Inner and Outer Product of the Wavefunctions
    • Quantum Physics
Replies
8
Views
2K
Determine a basis for kernel/nullspace
    • Calculus and Beyond Homework Help
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top