What Does Column n-Tuple Mean in Vector Spaces?

[vanckzhu]

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.

 

[Office_Shredder]

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).

 

[HallsofIvy]

A "column n-tuple" is simply an n-tuple written as a column. A column "3-tuple" would be somthing like
$$\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}$$
where a row "3-tuple" would be
$$\begin{bmatrix}1 & 2 & 3\end{bmatrix}$$.

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.

 

[vanckzhu]

A "column n-tuple" is simply an n-tuple written as a column. A column "3-tuple" would be somthing like
$$\begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}$$
where a row "3-tuple" would be
$$\begin{bmatrix}1 & 2 & 3\end{bmatrix}$$.

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 :).

 

[blue_raver22]

A column n-tuple is a vector in n-dimensional space, where each element of the vector represents a real number. In the context of the instructor's notes, V = R^n refers to the vector space of all possible column n-tuples of real numbers. So for R^3, an example of a column n-tuple would be (1,2,3)^T, where the superscript T denotes the transpose operation. This means that the vector is written as a column rather than a row.

As for online resources on dual spaces, there are many resources available, such as online lectures, tutorials, and textbooks. Some recommendations include Khan Academy, MIT OpenCourseWare, and YouTube channels such as 3Blue1Brown and MathTheBeautiful. It may also be helpful to consult with your instructor or classmates for further clarification on the notes.