What Does Column n-Tuple Mean in Vector Spaces?

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Discussion Overview

The discussion centers on the meaning of "column n-tuple" in the context of vector spaces, specifically within the framework of R^n. Participants explore the definition and representation of n-tuples, particularly in relation to matrices and their operations.

Discussion Character

  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant seeks clarification on the term "n-tuple," questioning whether a transpose of a tuple is an appropriate example for R^3.
  • Another participant confirms that an n-tuple is a list of n numbers, providing examples of 3-tuples and 4-tuples.
  • Several participants explain that a "column n-tuple" refers to an n-tuple represented as a column vector, distinguishing it from a row vector. They emphasize the importance of this distinction in matrix operations.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and representations of n-tuples and their significance in matrix operations. However, the initial question regarding the example of a transpose remains somewhat open, as it was not directly addressed by all participants.

Contextual Notes

The discussion does not resolve the potential confusion regarding the use of transposes in relation to n-tuples, leaving some assumptions about notation and context unaddressed.

vanckzhu
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Yet another silly question from me :/. From an instructor's notes: "Let V = R^n be the vector space of column n-tuples of real numbers." Not quite sure what the "tuple" means. For R^3, would an example be the transpose of (1,2,3)?

Thanks in advance for responding. While we're at it, any recommendations for online resources regarding dual spaces? These notes are a bit confusing.
 
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Yes, that's exactly what it is. An n-tuple is just a list of n numbers. So a 3-tuple would be something of the form (x,y,z) and a 4-tuple would be something of the form (w,x,y,z).
 
A "column n-tuple" is simply an n-tuple written as a column. A column "3-tuple" would be somthing like
[tex]\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}[/tex]
where a row "3-tuple" would be
[tex]\begin{bmatrix}1 & 2 & 3\end{bmatrix}[/tex].

The distinction is important when working with matrices. If A is an n by n matrix then "Av" is defined for v a column n-tuple while "vA" is define for v a row n-tuple.
 
HallsofIvy said:
A "column n-tuple" is simply an n-tuple written as a column. A column "3-tuple" would be somthing like
[tex]\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}[/tex]
where a row "3-tuple" would be
[tex]\begin{bmatrix}1 & 2 & 3\end{bmatrix}[/tex].

The distinction is important when working with matrices. If A is an n by n matrix then "Av" is defined for v a column n-tuple while "vA" is define for v a row n-tuple.

Thanks for the detailed explanation :).
 

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