Linear Algebra Exercises-n-Answers.pdf

148 Linear Algebra, by Hefferon
Topic: Geometry of Linear Maps
1 (a) To represent H, recall that rotation counterclockwise by θ radians is represented with respect
to the standard basis in this way.
( )
cos θ − sin θ
Rep E2 ,E 2
(h) =
sin θ cos θ
A clockwise angle is the negative of a counterclockwise one.
( ) ( √ √ )
cos(−π/4) − sin(−π/4) 2/2 2/2
Rep E2,E 2
(h) =
=
sin(−π/4) cos(−π/4) − √ √
2/2 2/2
This Gauss-Jordan reduction
(√ √ )
ρ 1+ρ 2 2/2 2/2 (2/ √ ( ) ( )
2)ρ
−→ √ 1 1 1 −ρ 2+ρ 1 1 0
−→
−→
0 2 (1/ √ 2)ρ 2
0 1 0 1
produces the identity matrix so there is no need for column-swapping operations to end with a
partial-identity.
(b) The reduction is expressed in matrix multiplication as
( ) ( √ ) ( )
1 −1 2/ 2 0
0 1 0 1/ √ 1 0
H = I
2 1 1
(note that composition of the Gaussian operations is performed from right to left).
(c) Taking inverses
( ) (√ ) ( )
1 0 2/2 0
H =
√ 1 1
I
−1 1 0 2 0 1
} {{ }
P
gives the desired factorization of H (here, the partial identity is I, and Q is trivial, that is, it is also
an identity matrix).
(d) Reading the composition from right to left (and ignoring the identity matrices as trivial) gives
that H has the same effect as first performing this skew
( ( )
x x + y
y)
⃗u
⃗v
↦→
y
−→
h(⃗u)
h(⃗v)
followed by a dilation that multiplies all first components by √ 2/2 (this is a “shrink” in that
√
2/2 ≈ 0.707) and all second components by
√
2, followed by another skew.
⃗u
⃗v
(
x
y)
↦→
( )
x
−x + y
−→
h(⃗u)
h(⃗v)
For instance, the effect of H on the unit vector whose angle with the x-axis is π/3 is this.