Why is b considered an element of R^m in the Ax=b theorem?

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Discussion Overview

The discussion revolves around the Ax=b theorem, specifically addressing why the vector b is considered an element of R^m. Participants explore the implications of matrix dimensions and the nature of vector multiplication in this context.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant notes that the theorem states b is in R^m because b is an mx1 column matrix, implying it has m rows.
  • Another participant explains that since A is an m x n matrix, it has m rows and n columns, and thus, for multiplication with a vector v, v must have n components, resulting in a product that is a vector with m components.
  • A participant seeks clarification on the notation used, questioning the expression "(mxn)(mx1)=(mx1)" and suggesting that the correct multiplication should involve an nx1 vector instead.
  • There is a reiteration of the matrix multiplication rules, with one participant asserting that a matrix with n columns and m rows multiplied by a matrix with 1 column and n rows results in a matrix with 1 column and n rows.

Areas of Agreement / Disagreement

Participants express differing interpretations of matrix multiplication notation and its implications for the dimensions of b. There is no consensus on the correct formulation or understanding of the multiplication process.

Contextual Notes

Some assumptions about matrix dimensions and the nature of vector multiplication remain unresolved, leading to confusion regarding the notation used in the discussion.

bonfire09
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In this theorem it states " Let A be a m x n matrix. That is For each vector b in R^m, the column Ax=b has a solution..." Why do they say that bεR^m? Is that because b is a mx1 column matrix where it has m rows making it belong to R^m?
 
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If A is an "m by n matrix" then it has n columns and m rows. That means that to multiply it by vector v, written as a column matrix, v must have n components, and then the product with be a vector, again written as a column matrix, will have m components. That is, A is from Rn to Rm.
 
so what your saying is that (mxn)(mx1)=(mx1).
and that means that bεR^m?
 
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bonfire09 said:
so what your saying is that (mxn)(mx1)=(mx1).
and that means that bεR^m?
I'm not sure I know what you mean by "(mxn)(mx1)= (mx1)". I thought we were talking about matrices, not numbers. What I said before was that a matrix with n columns and m rows, multiplied by a matrix with 1 column and n rows, gives a matrix with 1 column and n rows.

If I understand you notation, that would be "(mxn)(nx1)= (mx1)".
 

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