Confusion regarding dot product of vectors(row matrices)

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Discussion Overview

The discussion revolves around the dot product of vectors represented as row matrices and the implications of matrix multiplication rules in linear algebra. Participants explore the relationship between vector notation and matrix representation, addressing potential discrepancies and conceptual confusions regarding the existence of the dot product in different contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses confusion about the equivalence of a 1x3 row matrix and its vector notation, questioning why discrepancies arise in calculating the dot product.
  • Another participant clarifies that the dot product for matrices is technically an inner product, requiring one vector to be transposed, thus suggesting the notation xxT for the inner product.
  • A different participant references a textbook question about the possibility of a negative value for x.x, concluding that it cannot exist, indicating a misunderstanding of the context in which the dot product is defined.
  • Further clarification is provided that for a vector considered as a vector, the expression x.x is valid, but when treated as a matrix, the product must be xxT.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the dot product in the context of matrix representation versus vector notation. There are competing views on whether x.x can be defined in both contexts, leading to ongoing confusion.

Contextual Notes

Participants highlight limitations in understanding the definitions and contexts of vectors and matrices, particularly regarding the conditions under which the dot product is applicable. There is also mention of a specific textbook exercise that may not align with the participants' interpretations.

Who May Find This Useful

Readers interested in linear algebra, particularly those grappling with the concepts of vector representation, matrix multiplication, and the properties of dot products.

mdnazmulh
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I’ve got a confusion. We know a 1x3 row matrix is a 3-vector i.e.
x= [ a b c]
Matrix x can be written in vector notation like x= a i + b j + c k
where i, j, k are unit vectors along x,y & z axes.
For dot product of
x.x = a2 + b2 + c2 when x= a i + b j + c k

But according to the matrix multiplication rule, multiplication of two matrices is possible only when column of 1st matrix = row of the 2nd matrix.
So x.x = [ a b c] [ a b c] is not possible
My questions are :
(1) Both x= [ a b c] and x= a i + b j + c k are same vector.
Then why this discrepancy happens?
(2) Does really x.x exist when x = [ a b c]? Can we approach in any other way to define x.x when x = [ a b c] ?
I’m novice at linear algebra. So it would be helpful for me if you can explain elaborately. I’m really at a loss about that confusion.
 
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mdnazmulh said:
For dot product of
x.x = a2 + b2 + c2 when x= a i + b j + c k

But according to the matrix multiplication rule, multiplication of two matrices is possible only when column of 1st matrix = row of the 2nd matrix.

Hi mdnazmulh! :smile:

Technically, the product for "matrix" vectors is an inner product, and one of the vectors must be a transpose vector (written as a column vector instead of a row vector, or abbreviated xT).

So the inner product is xxT.

For details, you could see http://en.wikipedia.org/wiki/Inner_product_space" …

but I wouldn't bother until your professor deals with it. :smile:
 
Last edited by a moderator:
Thanks for your reply. Actually in the book of introductory linear algebra by Bernard Kolman there is a question that if x is an n-vector then is it possible that x.x can have negative value? And part (b) of the question says that if x.x=0 , what is x=0?
Now I understand the author placed those questions in the exercise just to check our conception. Answer MUST BE that x.x is no way possible.
Thank u again
 
mdnazmulh said:
… Now I understand the author placed those questions in the exercise just to check our conception. Answer MUST BE that x.x is no way possible.

No, that's not what I meant.

For a vector that's just a vector, x.x is correct.

Only if a vector is considered as a matrix (which I called '"matrix" vectors'), does the product have to be xxT …

that's what you originally asked about. :wink:
 

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