Describe how a 2x2 matrix can be used to represent a reflection transformation across the x-axis. Provide the matrix for this transformation.

4.12 Linear Transformations and Matrices - Review

Explain how a 2x2 matrix can be used to represent a rotation transformation in the plane. Provide an example with a specific angle of rotation.

How can a 2x2 matrix be used to represent a dilation transformation? Provide an example with a specific scaling factor.

Explain how a 2x2 matrix can be used to represent a shear transformation. Provide an example with a specific shear factor.

How can you determine if a given 2x2 matrix represents a linear transformation that preserves the area of geometric shapes? Provide an example.

Given the matrix $A = (\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix})$ , describe the effect of this matrix on the unit square with vertices at $(0, 0)$ , $(1, 0)$ , $(0, 1)$ , and $(1, 1)$ .

Explain how the matrix $B = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix})$ transforms a point $(x, y)$ in the plane.

Describe the effect of the matrix $C = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ on a line given by the equation $y = m x + b$ .

Given the matrix $D = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ , describe its effect on a shape in the plane.

10.

Explain how the matrix $E = (\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})$ transforms a point $(x, y)$ in the plane.

11.

Explain how a linear transformation can be used to rotate a vector in a 2D plane. Provide the transformation matrix for a counterclockwise rotation by an angle $θ$ .

12.

Describe how linear transformations can be used to scale an object in computer graphics. Provide the transformation matrix for scaling an object by a factor of 2 in the x-direction and 3 in the y-direction.

13.

How can linear transformations be used to reflect a vector across the y-axis? Provide the transformation matrix for this reflection.

14.

Explain how linear transformations can be used to shear an object in the x-direction. Provide the transformation matrix for shearing an object by a factor of 2 in the x-direction.

15.

Describe how linear transformations can be used to translate an object in a 2D plane. Why is translation not considered a linear transformation in the strict mathematical sense?

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AI InstructorAUG 12, 2024, 1:09:14 PM

A 2x2 matrix can represent a reflection transformation across the x-axis by using the following form:
$(\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$
This matrix, when multiplied by a vector, reflects the vector across the x-axis by negating the y-coordinate while keeping the x-coordinate unchanged.

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Which of the following matrices represents a reflection across the y-axis?

Select one of the following options:

A.
$(\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix})$
B.
$(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$
C.
$(\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$
D.
$(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$

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