What is a very simple example of a matrix acting on a vector space? 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?
 A: 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.  
A: 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
A: 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]
A: A vector space $V$ is a set equipped with addition $+$ operation, and also scalar multiplication by real (complex) numbers and they are distributive with respect to each other (see here for more detailed definition). The members of this set are rather abstractly called "vectors." A nontrivial example of such spaces that are very much used in physics and engineering is the set of real-valued functions $f:[a,b]\to \mathbb{R}$ (such as $\sin(t), t^2,...$). Note that one can define sum of two functions by pointwise adding them and define scalar multiplication of a function by a real number, by pointwise multiplying the function with the given scalar. 
Now, an $n\times m$ real-valued matrix $A$ acting on a vector space would be exactly similar to matrix acting on real numbers: given any $m$ vectors $v_1,\ldots,v_m$ of $V$ and stacking them on a vector (of vectors) $v=(v_1,\ldots,v_m)^T$, we can define $u=Av$, which is better described by a simple example. For this example, let $V$ to be the vector space of functions $f:[0,1]\to \mathbb{R}$ and let $v_1=t$ and let $v_2=\sin(t)$. Then:
$$
\begin{pmatrix}
1&2\\
-1&-1
\end{pmatrix}\begin{pmatrix}
t\\\sin(t)
\end{pmatrix}=\begin{pmatrix}t+2\sin(t)\\ -t-\sin(t)
\end{pmatrix}.
$$
In general, for $v\in V^m$, we can define $u=Av$ as a vector (of vectors) $u\in V^n$, given by: 
$$u_i=\sum_{j=1}^mA_{ij}v_j.$$
Note that since $V$ is a vector space (over reals) $A_{ij}v_j$ is defined and hence, $\sum_{j=1}^mA_{ij}v_j$ is also defined and would be in $V$.  
A: 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:


*

*MIT's Opencourseware on Linear Algebra

*Download for A First Course in Linear Algebra, by David Beezer

*Paul's Online Math Notes for Linear Algebra

