University of Pretoria Linear Algebra Worksheet

1. Let U , V , and W be vector spaces. Let L : V → W , L0 : V → W , andM : U → V be linear transformations.
(a) Show that L + L0 , defined by (L + L0 )(v) = L(v) + L0 (v) for any
vector v in V , is a linear transformationfrm V to W .
(b) Show that L ◦ M is a linear transformation from U to W .
(c) Explain why L + M and L ◦ L0 are NOT linear transformations in
general. What conditions could you put on U , V and W so that they
would be linear transformations?
√ 
 
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3
2. Let ~u1 =
and let ~u2 = √ .
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3
(a) Calculate the matrix A1 such that A1 ~x is the reflection of the vector
~x in R2 through the line spanned by ~u1 , i.e. A1 is the matrix which
represents the reflection in the standard basis.
(b) Calculate the matrix A2 such that A2 ~x is the reflection of the vector
~x in R2 through the line spanned by ~u2 , i.e. A2 is the matrix which
represents the reflection in the standard basis.
(c) Recall that the counterclockwise rotation of vectors in R2 by angleθ
cos θ − sin θ
(0 ≤ θ < 2π) is represented by the matrix R(θ) = . sin θ cos θ Show that A2 A1 represents a rotation, and identify the angle θ so that A2 A1 = R(θ). 3. Consider the plane P given by the equation x − 2y + z = 0. Let L : R3 → R3 be the linear transformation that reflects vectors through the plane P. (a) Find an orthonormal basis B of R3 consisting of eigenvectors for L. (b) Find the matrix [L]B B which represents L in the basis B. (c) Find the transition matrices TEB and TBE between the basis B and the  1 standard basis E consisting of the standard basis vectors ~e1 = 0, 0     0 0 ~e2 = 1 and ~e3 = 0. 0 1 (d) Find the matrix [L]EE which represents L in the standard basis. 4. We consider the same plane P from the previous problem and we use the same orthonormal basis B consisting of eigenvectors of L. Now consider a different linear transformation M : R3 → R3 which is rotation about the axis perpendicular to P by an angle of 90◦ counterclockwise when viewed from above. In other words, any vector in the direction of the axis is unchanged by M , while vectors in the plane P are rotated 90◦ about the origin in the plane and the rotation appears counterclockwise if you were looking down from (0, 0, +∞). (a) Find the matrix [M ]B B which represents M in the basis B. (b) Find the matrix [M ]EE which represents M in the standard basis. Round off matrix entries to three decimal places.