Applying a 3x3 matrix to a set of 4 coordinates

Hi, I'm struggling with this question on a practice exam. 

I have to apply the product of two 3x3 matrices, to a set of 4 coordinates for a square and explain the transformation. Putting the coordinates into a matrix gives me a 2x4 matrix. From my understanding, a 3x3 matrix cannot be applied to a 2x4 matrix, so I'm stuck on how to proceed. 

1 Answer

I am guessing that you are applying a linear transformation to map the 4 corners of a square. The linear transformation is a $3 \times 3$ matrix, say $A$. Coordinates of the square, $P_1, P_2, P_3, P_4$ are points in $\mathbb{R}^3$. In order to find the transformation of the square, you need to find transformation of the corners. Indeed you need to compute
\[PAP_i,   i=1,2,3,4\]
which are the transformations of the corners of the square under the map $A$. Note that the above matrix multiplication makes sense, and we are applying a $3 \times 3$ matrix, by a vector in $\mathbb{R}^3$. 

Erdos Erdos
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