Linear Transformation in ! 2 and ! 3 1. Find the matrix representation of the following transformations from L ! 2 ,! 2 with respect to the standard bases. If the transformation is not (
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linear, say why. Use Geogebra to verify your results. a. Scaling by a factor of k. b. Rotating by an angle of θ . c. Translating by (a,b) d. Reflecting over the y = mx line. e. Projecting onto the y = mx line. 2. The linear transformations L ! 2 ,! 2 form a vector space. Find a basis for (
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the vector space and describe the transformation induced by each of the basis vectors. 3. Suppose someone says that the linear transformations L ! 2 ,! 2 consist of (
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reflections, scalings, projections, rotations, and combinations thereof. Is that person correct? In what sense? Is there a basis that justifies this statement? 4. Find the matrix representation for the following linear transformations from L ! 3 ,! 3 with respect to the standard bases. (
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a. Rotation about the line x, y, z = at,bt,ct b. Projection onto the plane ax + by + cz = 0 c. Reflection across the plane ax + by + cz = 0