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?

√

1

3

2. Let ~u1 =

and let ~u2 = √ .

1

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.

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