What is an inner product
inner and outer product - definition
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.
For everyone who wants to help me (automatically generated by OnlineMathe):
"I want to create the solution in collaboration with others."
Online exercises (exercises) at unterricht.de:
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.
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?
- 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?
9:41 pm, March 31, 2016
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.
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?
10:19 p.m., March 31, 2016
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.
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