Link between geometric vector(which i knew in school) and vector as a matrix? What's the link between a vector (a one dimensional matrix) and a geometric vector (a line representing magnitude and direction)?I know that matrix is just a rectangular arrey and my second question is that how this arrey(just an arrey of number ) represents a physical quantity "Vector"(which has a direction and magnitude)?
 A: For dimensions 2 and 3 what is convenient about a bunch of numbers (array) is that it is easy to define linear operations which correspond to geometric actions.
Take for example translation and rotation, we can add vectors together and the geometric interpretation is concatenation of the "arrows" which point out a location.
Much of the convenience stems from the way matrix multiplication works.
Let us pretend we have a column vector with two elements: $a = \left[\begin{array}{r}a_x\\a_y\end{array}\right]$
The following operations can be applied by matrix multiplication from left:
First out is a rotation matrix that will rotate each vector around the middle point, origo $(0,0)$:
$$R_v = \left[\begin{array}{rr}\cos(v)&\sin(v)\\-\sin(v)&\cos(v)\end{array}\right]$$
This projection will calculate which part of the vector is in the x-direction:
$$M = \left[\begin{array}{rr}1&0\\0&0\end{array}\right]$$
This matrix will find the reflection as if we had a mirror along the x-axis:
$$M = \left[\begin{array}{rr}1&0\\0&-1\end{array}\right]$$
Some of the most convenient properties,

*

*We can switch coordinate systems for our vectors by mechanical calculation.

*We can concatenate operations ( since they are linear ) with matrix multiplication.

*We can solve equation systems asking what point to apply a geometric operation on a to get a given result.

A: They are essentially the same thing.
If you're talking about 2D space, then the vector $\vec{x} = (3, 4)$, for example, represents a geometric vector that, on a Cartesian plane, would point three units to the right and four units upwards (e.g. from the origin to the point (3, 4)). (I'm writing the vector as an ordered pair rather than as a 2x1 or 1x2 matrix, but under certain circumstances there's not a huge difference between those representations.)
If you're familiar with vector addition in the geometric sense (line them tip-to-tail and draw the arrow between them), then that exactly corresponds with co-ordinate-wise addition of the matrix-style vectors - so if you take the geometric vectors corresponding to (3, 4) and (2, 0) and add them, you get a vector that corresponds to (5, 4).
The matrix representation of a vector just happens to extend a bit better in certain fashions to more general kinds of vectors and vector spaces, where Cartesian geometry isn't necessarily the norm. For example, the space of polynomial functions is a vector space of infinite dimension.
A: Suppose you have a vector in the Euclidean plane with Cartesian coordinate system between points $P=(P_x,P_y)$ and $Q=(Q_x, Q_y)$. You can move that vector to the orgin, by taking points $(0,0)$ and $(Q_x-P_x, Q_y-P_y)$. Such a vector can be represented with just two numbers (we are in a 2-dimensional space), whoch can be aranged in a $2\times1$ matrix as follows: $$\left[\begin{array}{c}Q_x-P_x\\Q_y-P_y\end{array}\right]$$
In $\mathbb{R}^3$ it would be 3 numbers. With this matrix representation you can also have sums of vectors, dot products, etc.
I hope this helps $\ddot\smile$
