Determine the type of transformation represented by the following matrix. $$ \left[\begin{array}{cc}1 & 0 \ -2 & 1\end{array} ight] $$ Shear $$ \begin{array}{l}\checkmark ext { Select } \ ext { Upward } \ ext { Left } \ ext { Right } \ ext { Downward }\end{array} $$

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Determine the type of transformation represented by the following matrix. $$ \left[\begin{array}{cc}1 & 0 \ -2 & 1\end{array}ight] $$ Shear $$ \begin{array}{l}\checkmark 	ext { Select } \ 	ext { Upward } \ 	ext { Left } \ 	ext { Right } \ 	ext { Downward }\end{array} $$

UpStudy AI · Accepted Answer

We start with the matrix 
$$
\begin{pmatrix} 1 & 0 \ -2 & 1 \end{pmatrix}.
$$
A general point $$(x,y)$$ is transformed as follows:
$$
\begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \ -2 & 1 \end{pmatrix}\begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} x \ -2x+y \end{pmatrix}.
$$

This shows that
$$
x' = x \quad \text{and} \quad y' = y-2x.
$$
Because the $$x$$-coordinate is unchanged, the line $$x=0$$ is invariant. In other words, every point on the $$y$$-axis remains fixed:
$$
(0,y) \to (0, y-2\cdot 0) = (0,y).
$$

Since points not on the $$y$$-axis are displaced vertically by an amount proportional to their $$x$$-coordinate, this is a shear transformation. In standard terminology, the shear is described as *vertical* (because the displacement occurs in the $$y$$-direction) with the $$y$$-axis as the invariant line. Notice also that for points with positive $$x$$ (for example, $$(1,0)$$ maps to $$(1, -2)$$), the displacement is downward.

Thus, the transformation represented by the matrix is a shear transformation, specifically a vertical shear where the shear is downward for points with positive $$x$$.

$$
\boxed{\text{Vertical shear — Downward for } x>0.}
$$
The transformation is a vertical shear that moves points downward when $$ x $$ is positive.

Determine the type of transformation represented by the following matrix.

Shear

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Determine the type of transformation represented by the following matrix. Shear