Orthogonality and Orthonormality: Take 2

In summary: I appreciate the feedback.In summary, the given set of vectors in ℂ^3 is an orthogonal set, an orthonormal set, and an orthonormal basis. This is determined by using dot product algebra and calculating the inner product of each vector with itself, taking into account the special element of the inner product for complex vectors. Furthermore, this set spans the space of C3, making it a basis for the space.
  • #1
RJLiberator
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Homework Statement



Which of the following sets of vectors in ℂ^3 is an orthogonal set? Which is an orthonormal set? Which is an orthonormal basis?

[tex] \begin{pmatrix}
1/\sqrt{2}\\
0\\
1/\sqrt{2}
\end{pmatrix},
\begin{pmatrix}
-1/\sqrt{2}\\
0\\
1/\sqrt{2}
\end{pmatrix},
\begin{pmatrix}
0\\
(1+i)/\sqrt{2}\\
0
\end{pmatrix}[/tex]

Homework Equations

The Attempt at a Solution



Question 1: [/B] If I can conclude that the set is not orthogonal, then I can automatically conclude that the set is not orthonormal.
My answer: YES.

Question 2:
If I can conclude that the set is orthogonal, but not orthonormal, then I can conclude that the set is not an orthonormal basis.
My Answer: YES

I want to get these two understandings out of the way first. These are by definition, is this correct?Next, My work for this particular set:

If we let the first vector = v, second vecotr = w, third vector = z. By simple dot product algebra
v*w=0
v*z=0
w*z = 0

So we conclude that this set is orthogonal.

Now, we check for orthonormality by taking each vector and inner product it with itself. The interesting note here is that when you take the inner produce of z and itself you must use the complex conjugate of z and z itself. so <z*|z> which equals 1, instead of <z|z> which equals -i.

Here, in using that special element of the inner product we conclude that all 3 vectors inner producted with themselves is indeed 1 and so we see orthonormality.

Now, for orthonormal basis, we conclude that is IS an orthonormal basis considering there is a theorem that states that any orthonormal set of N vectors in V is an orthonormal basis for V. Since this set has 3 vectors and it is in ℂ ^ 3 we conclude that this is an orthonormal basis as well.
Is my analysis spot on? I feel like I am starting to finally get the hang of it.
 
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  • #2
I'd say this is spot on.

You nicely illustrated the philosophy behind your reasoning, invoked the right theorems.
One thing I would add regarding using the complex conjugate is that you need this to define a well-behaved norm. (i.e. recall the definitions you used and argue how to satisfy this for a complex vector)
And perhaps how this norm relates to cartesian norms as opposed to for example ##||z||_1 = \sum_{i=1}^3 |z_i|## (Taxicab or Manhattan norm).
 
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  • #3
Thank you kindly for the look over and the words of further advice.
:D
 
  • #4
Since you only provide ONE set of vectors, I assume your question is only about it. ie is it orthogonal, orthonormal and would it form a basis for C3. Since it spans the space (that is, every vector in C3 can be expressed in terms of it) it is all 3. (I actually am trusting you to have done the math right, looks right to me by inspection).
 
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  • #5
Indeed, I do not want to bore physics forums with more sets, I wanted to analyze this one as an example of my understanding.

Thank you ogg for your check over as well.
 

Related to Orthogonality and Orthonormality: Take 2

1. What is the difference between orthogonality and orthonormality?

Orthogonality refers to the property of two vectors being perpendicular to each other, meaning they form a 90 degree angle. Orthonormality takes this concept a step further and also requires the vectors to have a unit length of 1.

2. How does orthogonality play a role in linear algebra?

Orthogonality is an important concept in linear algebra as it allows for the decomposition of a vector into orthogonal components. This is useful in many applications such as solving systems of linear equations and performing transformations.

3. Can a set of vectors be both orthogonal and orthonormal?

Yes, a set of vectors can be both orthogonal and orthonormal. This means that they are not only perpendicular to each other, but also have a unit length of 1.

4. How is orthonormality helpful in matrix operations?

Orthonormal matrices have many useful properties in matrix operations. For example, they are easy to invert and their transpose is equal to their inverse. They also preserve the lengths and angles of vectors, making certain calculations easier and more efficient.

5. Can you provide an example of an orthonormal set of vectors?

One example of an orthonormal set of vectors is the standard basis vectors in 3-dimensional space: (1,0,0), (0,1,0), and (0,0,1). These vectors are not only perpendicular to each other, but also have a unit length of 1.

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