Set of m-vectors

Question

In my book (matrix computations by Gill) there's a term used quite a lot which I don't understand/find what is its meaning.

The set of all $m$-vectors that are linear combinations s of the columns of the $m \times n$ matrix $A$ is called range space, column space or simply the range of $A$.

Does "m-vectors" refer to vectors of dimension $m$?

Yes, I think it refers to a vector with $m$ coordinates – Dalamar Nov 15 '13 at 13:53 — Dalamar, Nov 15 '13 at 13:53

score 2 · Accepted Answer · answered Nov 15 '13 at 13:54

2

It will be elements of $\mathbb{R}^m$, i.e. vectors of dimension $m$.

See for example the definition on page 41 of Elementary Linear Algebra By Stewart Venit, Wayne Bishop, Jason Brown

answered Nov 15 '13 at 13:54

Henry

157,058

score 0 · Answer 2 · answered Nov 15 '13 at 13:57

Let $A$ be a $m \times n$ matrix, say $A = (v_1, v_2, . . . , v_n ) $ where $i$ th column of $A$ is $v_i$ and each $v_i$ is a vector in $\mathbb{F}^m$ . With this set up, by set of all $m-$vectors that are linear combinations of columns of $A$ they means :

set of all $ \alpha_1 v_1 + \alpha_2 v_2 +...+ \alpha_k v_k$ such that $\alpha_i \in \mathbb{F} $ and $1 \le k \le m$.

Set of m-vectors

2 Answers2