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Reading a book about quantum physics, the author described a matrix acting on a vector space. I have not really studied either matrices or vector spaces. But I want to start somewhere, so what is a very simple example of this type of mathematical situation -- a place for me to start?

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Actually the branch of Mathematics which study this kind of situations is Linear Algebra. There are several good books, but a good place to start from is wikipedia: en.wikipedia.org/wiki/Linear_algebra –  Giovanni De Gaetano Jun 15 '11 at 14:36
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This isn't quite as simple but the matrix $$\left[\matrix{\cos\theta & -\sin\theta \cr \sin\theta&\cos\theta}\right]$$ will rotate each vector in the xy-plane (which forms a vector space, the same space mentioned above, represent each point as an ordered pair (x,y) and define addition and scalar multiplication component-wise)

Try it out! pick a $\theta$ you can compute with easily like $\Large\frac{\pi}{2}$ and see where this matrix sends some points like (1,0) (1,1) [It should rotate them counter-clockwise]

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Here is a very elementary example. Consider the planar vectors with the usual operations of addition and scalar multiplication. This is a vector space.

Now consider the matrix $$\left[\matrix{1&0\cr 0&-1}\right]$$ The action on the vector $ai + bj$ by the matrix is defined as follows $$\left[\matrix{1&0\cr 0&-1}\right]\left[\matrix{a\cr b}\right]$$ to return $ai - bj$. Here, this matrix is reflecting the vector $ai + bj$ across the $x$-axis.

You can see how to reflect across th $y$-axis and the origin. The multiplication by the matrix defines transformations on the vector it acts upon.

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There are two demonstrations on Wolfram Demonstration Projects which may offer some intuition:

Linear Transformations of a Polygon and Linear Transformation Given by Images of Basis Vectors

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As other posters have said, the most familiar vector space to start in is the coordinate plane $\mathbb{R}^2$ and space $\mathbb{R}^3$. Your vectors are coordinate points, with vector addition and scalar multiplication being as you think (so $(a,b) + (c,d) = (a+c,b+d)$, and $k(a,b) = (ka,kb)$). Usually you present them vertically, like so: $\begin{bmatrix} a \\ b \end{bmatrix}$.

Matrices map vectors to vectors. Like functions, you can define a domain and range, and some (but not all) matrices have well defined inverses. The notation's usually $Ax=b$, where $A$ is a matrix, and $x$ and $b$ are vectors.

This is a (very basic) starting point, and there good references online, though I haven't used them myself. Here are a few:

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