Vector Analysis - Similarities on Orthonormal Basis

Recall that the length of a vector is the square root of its dot product with itself, and that dot products are preserved by linear transformations. So if you know the dot products of L(e1) and L(e2) with themselves, you know the length of L(ae1+be2) for any a,b, by linearity of L.
  • #1
math222
10
0

Homework Statement



Let L: R2 → Rn be a linear mapping. We call L a similarity if L stretches all vectors by the same factor. That is, for some δL, independent of v,

|L(v)| = δL * |v|

To check that |L(v)| = δL * |v| for all vectors v in principle involves an infinite number of calculations. Therefore a reduction to a finite number of checks can be useful for applications. Here is one such reduction.

Suppose that for the standard orthonormal basis {e1, e2} of R2, we have that L(e1) and L(e2) are orthogonal and have the same length. Show that L is a similarity. (What is δL?)

Homework Equations



Perhaps that orthogonal vectors have dot product zero? I'm not sure otherwise.

The Attempt at a Solution



So far I've been able to show that |L(e1)| = |L(e2)| = δL on the assumption that L is a similarity, but that doesn't really help that much. I'm not sure where to go beyond this.
 
Physics news on Phys.org
  • #2
math222 said:

Homework Statement



Let L: R2 → Rn be a linear mapping. We call L a similarity if L stretches all vectors by the same factor. That is, for some δL, independent of v,

|L(v)| = δL * |v|

To check that |L(v)| = δL * |v| for all vectors v in principle involves an infinite number of calculations. Therefore a reduction to a finite number of checks can be useful for applications. Here is one such reduction.

Suppose that for the standard orthonormal basis {e1, e2} of R2, we have that L(e1) and L(e2) are orthogonal and have the same length. Show that L is a similarity. (What is δL?)


Homework Equations



Perhaps that orthogonal vectors have dot product zero? I'm not sure otherwise.

The Attempt at a Solution



So far I've been able to show that |L(e1)| = |L(e2)| = δL on the assumption that L is a similarity, but that doesn't really help that much. I'm not sure where to go beyond this.

You need to use the orthogonal part. Any vector in R^2 can be written as a*e1+b*e2. What's L(a*e1+b*e2)? Compute its length using the dot product and that L(e1) is orthogonal the L(e2). It's really just the Pythagorean theorem.
 
  • #3
So do I assume that L(a*e1 + b*e2) is in fact similar?

And I'm not sure what you mean by using the dot product and orthogonality to compute the length. is there a formula I'm missing?
 
  • #4
math222 said:
So do I assume that L(a*e1 + b*e2) is in fact similar?

And I'm not sure what you mean by using the dot product and orthogonality to compute the length. is there a formula I'm missing?

No, you have PROVE it's similar. The length of a vector is ##\sqrt{v \cdot v}##. Is that the one you are missing? Use the linearity of L.
 
  • #5
ahhhh ok i'll try again, thanks for your help.
 
  • #6
ok got it. i basically ended up with a^2 + b^2. thanks again for your help.
 
  • #7
math222 said:
ok got it. i basically ended up with a^2 + b^2. thanks again for your help.

That's a little sketchy, but it sounds like you doing it right.
 

1. What is vector analysis?

Vector analysis is a mathematical tool used to study the properties and behavior of vectors, which are quantities that have both magnitude and direction. It involves performing operations such as addition, subtraction, and multiplication on vectors to understand their relationships and properties.

2. What is an orthonormal basis?

An orthonormal basis is a set of vectors that are both orthogonal and normalized. In other words, the vectors are perpendicular to each other and have a magnitude of 1. This basis is often used in vector analysis as it simplifies calculations and makes it easier to understand vector relationships.

3. How is vector analysis used in science?

Vector analysis is used in many areas of science, including physics, engineering, and computer science. It is particularly useful in studying motion, forces, and fields, as well as in solving problems involving multiple dimensions and directions.

4. What are the similarities between vectors on an orthonormal basis?

Vectors on an orthonormal basis share certain properties, such as having a magnitude of 1 and being perpendicular to each other. They also have similar relationships and can be added or subtracted in the same way. Additionally, they can be represented using the same coordinates or basis vectors.

5. Why is understanding vector analysis important for scientists?

Understanding vector analysis is important for scientists as it provides a powerful tool for describing and analyzing physical phenomena. It allows for the accurate representation and manipulation of quantities that have both magnitude and direction, making it essential in many fields of science.

Similar threads

  • Calculus and Beyond Homework Help
Replies
16
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
999
  • Linear and Abstract Algebra
Replies
2
Views
894
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
22
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
9
Views
152
Back
Top