What is an inner product

inner and outer product - definition

Hello,

I have a question about the inner product and the outer product:

1) I found two different approaches for the definition of the inner product:

... inner product =

... inner product =

-> are both definitions correct or is one wrong?

The inner product indicates, so to speak, the position (angle) of the Vekotren to one another. - That means if the inner product is 0, they should be right-angled!


2) I have no idea what that should be and how to interpret it. I always read about the outer product, but I can't understand what it is and how it differs from the inner product!

As an example I have here:




What is the rank of the outer product of these 2 vectors?

-> As long as I don't know exactly how the outer product is defined, I can't calculate it.


Thanks!
Kind regards

For everyone who wants to help me (automatically generated by OnlineMathe):
"I want to create the solution in collaboration with others."Suitable for this at OnlineMathe:

Online exercises (exercises) at unterricht.de:
According to the wiki, there is an outer product. 2 options:

1. it is the cross product

2. it is the dyadic product


Now you have to annoy your lecturer for what he means!

He has to give you an answer to that, otherwise we’ve made a complaint, the guy is gone now.
He writes exactly this question "Given the vectors a and b below. What is the rank of the outer product?

But isn't that the inner product?



For me the inner product would be:

But what exactly is the outer product? - Is that then or is that the cross product?

So I've now thought it through again!


- In this case the inner product must be, since only a number should come out, i.e. a scalar.

- the outer product has to be in this case, because a vector should come out ...

the result would be:


The rank of the outer product would be 3!


-> Am I correct here?
Sorry, the rank of the matrix would then be 1, because I still have to bring it into the staircase shape and the first row remains there and the other two fall away, right?

ledum

9:41 pm, March 31, 2016

Hello
the matrix and rank 1 are correct, you do not have to bring them to a stepped form, since all rows are multiples of, i.e. proportional.
what surprised me is that you write.
If vectors are called single-row or single-column matrices, then usually with a a row vector, with a column vector.
but the outer product with the result matrix is ​​column row, the inner product = scalar product is row times column
however, if you have given, column times row is really a matrix.
while is a scalar.
Greetings ledum
@Ledum

Thanks first of all!

So that with the rank is clear to me in the example, but would it be a not so simple example would I have to find out with the staircase shape (or "row reduced echolon form")? -> In addition a question, what exactly is the staircase shape or what exactly does it express in a system of equations?

To the point with the inner and outer product:

- So seen in this way it is more a matter of definition how the vector is given, or how? In principle, the inner product always results in a scalar, i.e. a number, and a vector in the outer product. - But the whole thing depends on whether a is given as a row vector / column vector and b as a row vector / column vector?

-> Do I understand correctly?

ledum

10:19 p.m., March 31, 2016

Hello column times line is always a matrix, line times column is a scalar, if you follow the rules of the matrix. used.
obviously one calls the first outer product in your lecture because the "normal (" line vector is outside = left.
(others call it tensor product)
if you connect 2 normal vectors with the outer product, the matrix is ​​always rank 1.
to determine the rank you can determine how many lin independent lines (or columns) you have, that works. . with the determination of the kernel that has here (with 2 zero lines that means that only 1 remains for the rank of the matrix. or you can see through the 2 zero lines that there is only 1 lin independent line vector.
Greetings ledum

Thanks for your detailed explanation! ;-)