98 lines
2.5 KiB
Text
98 lines
2.5 KiB
Text
#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
|
|
|
|
#show: ams-article.with(title: "Exam Solutions")
|
|
|
|
Q6.
|
|
|
|
The general clockwise rotation matrix in $RR^2$ is
|
|
|
|
$
|
|
mat(cos theta, sin theta; -sin theta, cos theta)
|
|
$
|
|
|
|
We have $theta = (3pi)/2$, and
|
|
|
|
$ cos((3pi) / 2) = 0, sin((3pi) / 2) = -1 $
|
|
|
|
So our particular rotation matrix is
|
|
|
|
$
|
|
T = mat(0, -1; 1, 0)
|
|
$
|
|
|
|
Clearly, the linear transformation that reflects a vector across the vertical
|
|
axis changes the first standard basis vector, $vec(1, 0)$, to $vec(-1, 0)$.
|
|
|
|
This corresponds to the linear transformation (matrix)
|
|
|
|
$
|
|
S = mat(-1, 0; 0, 1)
|
|
$
|
|
|
|
Matrix composition, $compose$, is an equivalent notion to matrix multiplication. Therefore, we have the two compositions.
|
|
|
|
+ #[
|
|
$T compose S$, the linear transformation corresponding to a reflection followed by rotation:
|
|
|
|
$
|
|
T compose S &= mat(0, -1; 1, 0) mat(-1, 0; 0, 1) \
|
|
&#[using _column by coordinate_ rule] \
|
|
&= mat((-1 vec(0,1) + 0 vec(-1,0)), (0 vec(0,1) + 1 vec(-1,0))) \
|
|
&= mat(0, -1,;-1, 0)
|
|
$
|
|
]
|
|
|
|
+ #[
|
|
$S compose T$, the linear transformation corresponding to rotation, followed by reflection. Since matrix composition is generally not commutative, we obtain a different matrix.
|
|
|
|
$
|
|
S compose T &= mat(-1,0; 0,1) mat(0,-1; 1,0) \
|
|
&#[using _column by coordinate_ rule] \
|
|
&= mat((0 vec(-1, 0) + 1 vec(0, 1)), (-1 vec(-1, 0) + 0 vec(0, -1))) \
|
|
&= mat(0, 1; 1, 0)
|
|
$
|
|
]
|
|
|
|
#pagebreak()
|
|
|
|
Q7.
|
|
|
|
#enum(
|
|
numbering: "7.1",
|
|
[
|
|
The matrix $A$ corresponds to a linear transformation
|
|
$T: RR^4 |-> RR^3$. $A$ has 3 rows and 4 columns, so its matrix-vector
|
|
multiplication is only defined when with vectors in $RR^4$. Accordingly, it
|
|
will output a vector in $RR^3$.
|
|
|
|
So, $p = 4$.
|
|
],
|
|
[
|
|
See above explanation. $q = 3$.
|
|
|
|
],
|
|
[
|
|
To find all vectors $arrow(x) in RR^4$ whose image under $T$ is $arrow(b)$, we seek all solutions $arrow(x) = (x_1, x_2, x_3, x_4, x_5)^T$ to the equation
|
|
$ T arrow(x) = arrow(b) $
|
|
|
|
We can do this using our usual row reduction methods.
|
|
|
|
$
|
|
mat(augment: #(-1), -2,3,7,-11,-3;1,0,-2,1,3; 1,-1,-3,4,2) \
|
|
mat(augment: #(-1), 1,-1,-3,4,2; 0, 1, 1, -3, 1; 0, 0, 0, 0, 0)
|
|
$
|
|
|
|
We now have the augmented matrix in echelon form. So, $x_4$ and $x_3$ are free. Then, let $s,t in RR$ be free variables
|
|
$
|
|
x_1 &= 3 - t + 2s \
|
|
x_2 &= 1 + 3t - s \
|
|
x_3 &= s \
|
|
x_4 &= t
|
|
$
|
|
So, all vectors $arrow(x)$ are of the form
|
|
|
|
$
|
|
arrow(x) = vec(3+2s - t, 1 - s + 3t, s, t)
|
|
$
|
|
],
|
|
)
|