0
$\begingroup$

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?

  • 1
    You cannot multiply $A$ with a vector in $\mathbb{R}^3$. Do you mean $\mathbb{R}^2$ instead?2012-01-28
  • 0
    Corrected. New to this.2012-01-28
  • 0
    The commas were neither corrected nor addressed. $\;$2012-01-28
  • 0
    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.2012-01-28

1 Answers 1