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 :).