Subjects linear algebra

Matrix Transformations

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Matrix Transformations


1. The problem shows two 3x3 matrices labeled (c) and (d). 2. Matrix (c) is: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 5 & 0 & 1 \end{pmatrix}$$ This is an example of an elementary matrix representing a shear transformation in the x-z plane because the entry 5 in the third row and first column adds 5 times the first row to the third row. 3. Matrix (d) is: $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ This matrix scales the y-coordinate by 5, because the second diagonal entry is 5 while the others are 1. 4. To summarize the transformations: - Matrix (c) applies a shear adding 5 times the x-component (first coordinate) to the z-component (third coordinate). - Matrix (d) applies a vertical scaling by a factor of 5 along the y-axis. 5. These matrices are used in linear algebra to represent different affine transformations in 3D space.