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Exercises related to linear transformations in geometry. It includes questions on showing that a given transformation is linear, finding a matrix that represents the transformation, computing the transformation using the matrix, and concluding whether the transformation is invertible. The document also includes exercises on the effect of linear transformations on given vectors.
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T
x 1 x 2 x 3
x 1 x 2 0
a) Show that T is a linear transformation.
b) Find a matrix A such that T (~x) = A~x for all ~x.
c) Compute T
(^) and T
, using both the definition of T and the matrix of transforma- tion A.
d) Use part c) to conclude that T is not an invertible transformation.
- 1
1
2
2
In each part below, you are given a matrix A. Draw what happens to the vectors ~e 1 and ~e 2 after applying the linear transformation T (~x) = A~x. Describe the effect of the linear transformation in words.
(a) A =
(b) A =
(a) A 2 × 2 matrix that rotates every vector in R^2 30 ◦^ counterclockwise and scales it by a factor of 2.
(b) A 3 × 3 matrix that reflects every vector in R^3 over the xy−plane.
(c) A 3 × 3 matrix that rotates every vector in R^3 180 ◦^ around the y−axis.