# Relationship between coordinate systems and linear transformations

Below is a shear transformation matrix A. If you multiply a vector in $\mathbb{R}^{2}$ by the matrix A you will get back a sheared vector in $\mathbb{R}^{2}$.

A = $\begin{bmatrix} 1&2 \\0&1 \end{bmatrix}$

However, the above matrix is also the inverse matrix (A is the inverse of B) of the following:

B = $\begin{bmatrix} 1&-2 \\0& 1 \end{bmatrix}$

If you multiply a vector (standard basis) by the matrix A you will get back the coordinates of that vector relative to the coordinate system defined by matrix B. Correct?

Therefore, a matrix's "meaning" is dependent on the context we use it in. Is that right?

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You cannot multiply $A$ with a vector in $\mathbb{R}^3$. Do you mean $\mathbb{R}^2$ instead? –  Jalaj Jan 28 '12 at 2:35
Corrected. New to this. –  koin Jan 28 '12 at 3:20
The commas were neither corrected nor addressed. $\;$ –  Ricky Demer Jan 28 '12 at 3:42
A matrix always represents a linear transformation. What is really changing is our interpretation of the elements of $\mathbb{R}^2$, as either locations on the plane, or as coordinate vectors that describe linear combinations of certain vectors. –  Arturo Magidin Jan 28 '12 at 4:24