TITLE: Significance of Matrix-Vector multiplication
QUESTION [8 upvotes]: Can someone give me an example illustrating physical significance of the matrix-vector multiplication? 

Does multiplying a vector by matrix transforms it in some way?
Do left & right multiplication signify two different things?
Is matrix a scalar thing? (EDIT)

Thank you.

REPLY [6 votes]: I believe that parts 2 and 3 of your question have been answered well. I'd like to take a stab at part 1, though the other answers to this part are probably better.
There's an interesting way of thinking of the application of a matrix to a vector using the Singular Value Decomposition (SVD) of a matrix. Let A be an $m \times n$ rectangular matrix. Then, the SVD of A is given by $A = U \Sigma V^T$, where $U$ is an $m \times m$ unitary matrix, $\Sigma$ is an $m \times n$ diagonal matrix of so-called singular values and $V$ is an $n \times n$ unitary matrix. For more on the SVD, check out the Wikipedia article: http://en.wikipedia.org/wiki/Singular_value_decomposition
That same article contains proof that every matrix has an SVD. Given that fact, we can now think of matrix vector multiplication in terms of the SVD. Let $\bf x$ be a vector of length $n$. We can write the matrix-vector multiplication as ${\bf b} = A {\bf x}$. But, $A{\bf x} = U\Sigma V^T {\bf x}$.
Since $V$ is unitary, $V {\bf x}$ does not change the magnitude of ${\bf x}$. Unitary matrices applied to a vector only change the direction of the vector (rotate it by some angle). The product $V^T {\bf x}$ rotates ${\bf x}$.
$\Sigma$ is a diagonal matrix. Its entries directly multiply the corresponding entries of the vector ${\bf x}$, thus scaling the vector (increasing its length) along the axis around which $V$ rotated ${\bf x}$. However, remember that the rotated vector $V^T{\bf x}$ is a vector of length $n$ while $\Sigma$ has dimensions $m \times n$. This means that $\Sigma$ is also embedding the vector $V^T{\bf x}$ in an $m$-dimensional space, i.e., changing the dimensions of the vector. If $m=3$ and $n=2$, for example, $\Sigma$ scales the 2D vector $V^T{\bf x}$ in 2 dimensions and then "places" it in a 3D space.
Finally, we have the product of the unitary matrix $U$ with the $m$-dimensional vector $\Sigma V^T{\bf x}$. $U$ rotates that vector in the $m$-dimensional space. 
Every matrix thus potentially rotates, scales and embeds and then again rotates a vector when applied to a vector. When $m=n$, of course, a matrix-vector product doesn't involve any embedding- simply a rotation, scaling and another rotation. Like a little assembly line.