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With regard to vectors, how is this (form 1): $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ Different to this (form 2): $$\begin{bmatrix}1\ 2\ -1 \end{bmatrix}$$
I would think that the first set consists of magnitudes of the same variable (where simply the sum gives the magnitude), while the second refers to the coefficients of three different variables.

In my book they take the magnitude of form 1 by $$\sqrt{1^2+2^2+(-1)^2}$$ That would indicate a set of vectors in different directions. So my question is; which of the two (rows or columns) give an indication of the variable or the vector direction / dimension?

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  • $\begingroup$ What is the difference between these: $\mathcal{7}$ $\mathbf{7}$ $7$ 7 $\endgroup$
    – vadim123
    May 15, 2015 at 14:13
  • $\begingroup$ @vadim123 C'mon, this is a valid question. A similar one might be Is “a+0i” in every way equal to just “a”?. $\endgroup$
    – user1729
    May 15, 2015 at 14:17
  • $\begingroup$ @user1729, my comment was not meant to denigrate the question, but to answer it. The difference is one of notation only. The link you provide is a substantially different question. $7$ is an element of both $\mathbb{Q}$ and $\mathbb{C}$, but $7+0i$ is an element of $\mathbb{C}$ only. $\endgroup$
    – vadim123
    May 15, 2015 at 14:24
  • $\begingroup$ You might also be interested in math.stackexchange.com/q/1198729/139123 and its answer, although I think you have some questions about the meanings of the individual numbers within a column vector that are not really considered there. $\endgroup$
    – David K
    May 16, 2015 at 4:04

1 Answer 1

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What is a vector? Well, it is not "something with direction and magintude". Instead a vector is an arbitrary element of a vector space. Of course this merely shifts the question to: What is a vector space? But that can be defined to be a structure with quite simple axioms.

That being said, triples of real numbers from a vector space (with suitable "obvious" operations), no matter which notation we use to write down sucxh a triple. However, while we'd usually consider $\mathbb R^3$ as the set of triples of real numbers and write such triples in the form $(x,y,z)$, for the vector space $\mathbb R^3$, it is customary to use the column form, i.e. write vectors as in your form 1. The reason is that this makes most sense when introducing matrix multiplication (i.e., $Av$ where $A$ is a matrix and $v$ a vector). That would make a row vector a one-by-three matrix, an element of $\mathbb R^{1\times 3}$. But these two structures are isomorphic (i.e., essentiually indistinguishable).

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  • $\begingroup$ Ok, well that changes the way I think about it. It seems the "direction & magnitude" idea has fully hardwired itself. How would we then indicate the difference between a vector space defined by a two dimensional plane of real numbers and and an infinite three dimensional space? $\endgroup$
    – Chris-Al
    May 15, 2015 at 14:29
  • $\begingroup$ How would I indicate the direction of an element in such a vector space? If I have two triples that are orthogonal, then how could that be possible if my entries do not correspond to some direction? Or, are you saying that they do, but just that the column / row notation are the same thing? $\endgroup$
    – Chris-Al
    May 15, 2015 at 15:10
  • $\begingroup$ @ChrisAl In a vector space (over the real numbers, to be sure) where we have a notion of orthogonality by means of some scalar product (not all vector spaces have such a scalar product naturally!), we can in fact define angles between nonzero vectors via $\cos(v,w)=\frac{v\cdot w}{\sqrt{v\cdot v}\sqrt{w\cdot w}}$ or simply say that two vectors point in the same direction if they are obtained from each other by multipying with a (positive) constant. $\endgroup$
    – Hagen von Eitzen
    May 15, 2015 at 16:31

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