Kindly explain in simple terms because I have limited knowledge of matrices. Also, what is a transformation matrix?
2026-03-29 05:27:23.1774762043
'For transformation matrices, the rank of matrix tells you the dimensions of the output.' What does this statement mean?
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A transformation matrix is just a matrix. The word transformation emphasises that the matrix can take you from a (vector) space with one dimension (say 3D space) into another one (say 2D space). Going from 3D space into 2D space requires a matrix of 3 columns of 2 numbers each. Each of these 3 columns is 2D and after applying the matrix to the column $(x, y, z)$ you get a 2D column. This resulting column of numbers is a linear combination of the columns of the original matrix $$\begin{bmatrix} 1& 3& 2 \\ 2& 1 & 5 \end{bmatrix}\begin{bmatrix} x\\ y\\ z \end{bmatrix}=x\begin{bmatrix} 1\\ 2 \end{bmatrix}+y\begin{bmatrix} 3\\ 12 \end{bmatrix}+z\begin{bmatrix} 2\\ 5 \end{bmatrix}$$ The columns of the matrix thus describe all possible outcomes of the transformation in 2D. All three of these columns cannot be independent of each other - we must be able to write one of them as a combination of the other 2. Sometimes we might be able to do more - maybe two of them are multiples of the other. The rank of the matrix is the number of independent columns and as the output in 2D is a combination of these columns its dimension must be this rank.