Practical Application of Linear Transformation as taught in Linear Algebra. Can anyone provide a basic practical use of linear transformation? ( Or maybe a metaphor like example) 
Also visual, practical, or real applications for 
one to one 
onto
null(A)
determinant 
rank
I'm a person who likes to picture big, and it's very annoying to motivate myself to focus on the details (such as the definitions and theorems) first when I don't see the real life use of the new concept. 
Visualizing also helps me so if the application is visual and practical, it would be great (and simple)
For example, our book talks about how matrices could be used to encode messages, and the decoder is the inverse of the matrix. This was fantastic, not only I internalized the concept but also now I have a new "tool" in my pocket.
Thanks.
 A: I realize that this may not be exactly what you are looking for, but I always found it a very nice simple example illustrating the usefulness of matrices.
Consider the following problem (with parameters that I made up): Every year people get the flu. Those who get the flu this year have a $30\%$ chance of getting the flu next year, whereas those who do not get the flu this year have a $40\%$ chance of getting the flu next year. This problem can be modelled by the matrix (why?)
$$
M=\left(\begin{matrix}0.3&0.4\\0.7&0.6\end{matrix}\right).
$$
Supposing that this year there were $x$ people who got the flu and $y$ people who did not get the flu, we can now give an estimate for the number of people, say $x_n$ who get the flu after $n$ years, and the number of people, say $y_n$, who do not get the flu, namely
$$
\left(\begin{matrix}x_n\\y_n\end{matrix}\right)=\left(\begin{matrix}0.3&0.4\\0.7&0.6\end{matrix}\right)^n\left(\begin{matrix}x\\y\end{matrix}\right).
$$
Now, matrix multiplication is a laborious process unless the matrices have a simple form, say a diagonal form for instance. Luckily, there is basis which diagonalizes this matrix, i.e. we can pick an invertible matrix $U$ such that
$$
\left(\begin{matrix}0.3&0.4\\0.7&0.6\end{matrix}\right)=U^{-1}\left(\begin{matrix}\lambda_1 &0\\0&\lambda_2\end{matrix}\right)U,
$$
and this is something you can show by methods relying on the determinant. It follows that
$$
\left(\begin{matrix}x_n\\y_n\end{matrix}\right)=U^{-1}\left(\begin{matrix}\lambda_1^n&0\\0&\lambda_2^n\end{matrix}\right)U\left(\begin{matrix}x\\y\end{matrix}\right).
$$
With this much simpler form, it is possible to answer questions such as whether or not the population 'stabilizes' in the sense that there are $x_\infty,y_\infty$ such that $x_n\rightarrow x_\infty,y_n\rightarrow y_\infty$, as one might expect.
