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Linear Transformations
Related Topics
Wize University Linear Algebra Textbook > Linear Transformations
Linear Transformations
5 Activities
Practice: Identifying Non-Linear Transformations
Select all of the following transformations that are
non-linear
.
A)
T
(
[
x
y
]
)
=
[
2
x
3
y
−
x
y
]
T\left(\begin{bmatrix} x\\ y\\ \end{bmatrix}\right) = \begin{bmatrix} 2x\\ 3y-x\\ y \end{bmatrix}
T
(
[
x
y
]
)
=
2
x
3
y
−
x
y
B)
S
(
[
x
y
]
)
=
[
2
x
2
3
x
y
+
7
]
S\left(\begin{bmatrix} x\\ y\\ \end{bmatrix}\right) = \begin{bmatrix} 2x^2\\ 3xy+7\\ \end{bmatrix}
S
(
[
x
y
]
)
=
[
2
x
2
3
x
y
+
7
]
C)
R
(
[
x
y
]
)
=
[
x
0
]
R\left(\begin{bmatrix} x\\ y\\ \end{bmatrix}\right) = \begin{bmatrix} x\\ 0\\ \end{bmatrix}
R
(
[
x
y
]
)
=
[
x
0
]
D)
P
(
[
x
y
z
]
)
=
[
−
1
2
y
x
−
y
]
P\left(\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix}\right) = \begin{bmatrix} -1\\ 2y\\ x-y\\ \end{bmatrix}
P
x
y
z
=
−
1
2
y
x
−
y
I don't know
Check Submission
More Linear Transformations Questions:
Consider the linear transformation
T
:
R
2
→
R
3
T:\mathbb{R}^2 \rightarrow \mathbb{R}^3
T
:
R
2
→
R
3
such that
T
[
1
2
]
=
[
1
2
3
]
T\begin{bmatrix} 1\\2 \end{bmatrix}=\begin{bmatrix} 1\\2\\3 \end{bmatrix}
T
[
1
2
]
=
1
2
3
and
T
[
1
1
]
=
[
−
2
−
1
2
]
T\begin{bmatrix} 1\\1 \end{bmatrix}=\begin{bmatrix} -2\\-1\\2 \end{bmatrix}
T
[
1
1
]
=
−
2
−
1
2
Then
T
[
1
−
1
]
T\begin{bmatrix} 1\\-1 \end{bmatrix}
T
[
1
−
1
]
is equal to
True or False: Let
T
:
R
4
→
R
2
T:\mathbb{R}^4\to\mathbb{R}^2
T
:
R
4
→
R
2
defined by
T
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
x
1
+
2
x
2
,
x
3
−
x
4
)
T(x_1,x_2,x_3,x_4)=(x_1+2x_2,x_3-x_4)
T
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
x
1
+
2
x
2
,
x
3
−
x
4
)
, then
T
T
T
is a linear transformation and the matrix of
T
T
T
is given by
T
=
[
1
2
0
0
0
0
1
−
1
]
T=\begin{bmatrix}1&2&0&0\\0&0&1&-1\end{bmatrix}
T
=
[
1
0
2
0
0
1
0
−
1
]
Determine if a transformation is linear
Let
W
=
{
(
x
,
y
,
z
)
∣
x
,
y
,
z
∈
R
+
}
W=\{(x,y,z)\;|\;x,y,z\in\mathbb{R}^+\}
W
=
{(
x
,
y
,
z
)
∣
x
,
y
,
z
∈
R
+
}
be the vector with defined operations:
(
x
1
,
y
1
,
z
2
)
+
(
x
2
,
y
2
,
z
2
)
=
(
x
1
x
2
,
y
1
y
2
,
z
1
z
2
)
(x_1,y_1,z_2)+(x_2,y_2,z_2)=(x_1x_2,\;y_1y_2,\;z_1z_2)
(
x
1
,
y
1
,
z
2
)
+
(
x
2
,
y
2
,
z
2
)
=
(
x
1
x
2
,
y
1
y
2
,
z
1
z
2
)
k
⋅
(
x
,
y
,
z
)
=
(
x
k
,
y
k
,
z
k
)
k\cdot(x,y,z)=(x^k,\;y^k,\;z^k)
k
⋅
(
x
,
y
,
z
)
=
(
x
k
,
y
k
,
z
k
)
Matrix of a Linear Transformation
Consider the map
T
:
R
3
→
R
3
x
⃗
→
p
r
o
j
(
1
,
1
,
0
)
x
⃗
+
2
p
r
o
j
(
1
,
0
,
−
1
)
x
⃗
−
p
r
o
j
(
0
,
0
,
1
)
x
⃗
\begin{array}{l}T:\mathbb{R}^3\rightarrow\mathbb{R}^3\\\vec x\rightarrow proj_{(1,1,0)}\vec x+2proj_{(1,0,-1)}\vec x-proj_{(0,0,1)}\vec x\end{array}
T
:
R
3
→
R
3
x
→
p
r
o
j
(
1
,
1
,
0
)
x
+
2
p
r
o
j
(
1
,
0
,
−
1
)
x
−
p
r
o
j
(
0
,
0
,
1
)
x
a)
Show that T is a linear transformation
b)
Compute the matrix of T, and its inverse matrix.
Find the matrix of a given transformation
(a)
Let
L
:
R
3
⟶
R
4
L:\mathbb{R}^3\longrightarrow\mathbb{R}^4
L
:
R
3
⟶
R
4
be defined by:
L
(
(
x
,
y
,
z
)
)
=
(
2
x
−
y
,
2
y
+
3
z
,
−
3
x
+
5
z
,
x
−
y
+
z
)
L\big((x,y,z)\big)=(2x-y,2y+3z,-3x+5z,x-y+z)
L
(
(
x
,
y
,
z
)
)
=
(
2
x
−
y
,
2
y
+
3
z
,
−
3
x
+
5
z
,
x
−
y
+
z
)
Find the matrix
A
A
A
that represents
L
L
L
Supposse
T
[
0
1
]
=
[
1
1
2
]
,
T
[
1
1
]
=
[
2
1
−
1
]
,
B
=
[
0
1
0
]
,
a
n
d
C
=
[
0
0
1
]
T\begin{bmatrix} 0\\1 \end{bmatrix}=\begin{bmatrix} 1\\1\\2 \end{bmatrix},\; T\begin{bmatrix} 1\\1 \end{bmatrix}=\begin{bmatrix} 2\\1\\-1 \end{bmatrix},\; B=\begin{bmatrix} 0\\1\\0 \end{bmatrix},\; and\; C=\begin{bmatrix} 0\\0\\1 \end{bmatrix}
T
[
0
1
]
=
1
1
2
,
T
[
1
1
]
=
2
1
−
1
,
B
=
0
1
0
,
an
d
C
=
0
0
1
Then:
Finding Non-linear Transformation
Which of the following transformations are linear?
a
)
[
x
y
]
→
[
x
−
y
2
x
+
y
+
6
0
]
a)\begin{bmatrix} x\\y \end{bmatrix}→\begin{bmatrix} x-y\\ 2x+y+6\\ 0 \end{bmatrix}
a
)
[
x
y
]
→
x
−
y
2
x
+
y
+
6
0
Let T be a linear transformation. Suppose that
T
(
[
1
0
1
]
)
=
[
−
2
0
−
1
]
,
T
(
[
0
1
1
]
)
=
[
0
3
2
]
,
T
(
[
0
1
0
]
)
=
[
0
−
4
0
]
T\left(\begin{bmatrix} 1\\0\\1 \end{bmatrix}\right)=\begin{bmatrix} -2\\0\\-1 \end{bmatrix},T\left(\begin{bmatrix} 0\\1\\1 \end{bmatrix}\right)=\begin{bmatrix} 0\\3\\2 \end{bmatrix},T\left(\begin{bmatrix} 0\\1\\0 \end{bmatrix}\right)=\begin{bmatrix} 0\\-4\\0 \end{bmatrix}
T
1
0
1
=
−
2
0
−
1
,
T
0
1
1
=
0
3
2
,
T
0
1
0
=
0
−
4
0
a
)
F
i
n
d
T
(
[
1
0
0
]
)
a)\ Find\ T\left(\begin{bmatrix} 1\\0\\0 \end{bmatrix}\right)
a
)
F
in
d
T
1
0
0
Find a linear transformation T from R
2
to R
2
, if
T
(
[
1
−
4
]
)
=
[
−
3
6
]
T\left(\begin{bmatrix} 1\\-4 \end{bmatrix}\right)=\begin{bmatrix} -3\\6 \end{bmatrix}
T
(
[
1
−
4
]
)
=
[
−
3
6
]
and it transforms points on line
y
=
0
y=0
y
=
0
to new line
y
=
2
x
y=2x
y
=
2
x
and vice versa:
Let
u
⃗
=
[
−
3
1
4
]
\vec{u} = \begin{bmatrix} -3 \\ 1 \\ 4 \end{bmatrix}
u
=
−
3
1
4
and let
T
:
R
3
→
R
3
T : \mathbb{R}^3 \rightarrow \mathbb{R}^3
T
:
R
3
→
R
3
be the linear transformation defined by
T
(
v
⃗
)
=
proj
u
⃗
(
v
⃗
)
T(\vec{v}) = \operatorname{proj}_{_{\vec{u}}} (\vec{v})
T
(
v
)
=
proj
u
(
v
)
. Find the matrix of
T
T
T
.
Let
T
:
R
2
→
R
2
T:\mathbb{R}^2\to\mathbb{R}^2
T
:
R
2
→
R
2
be a linear transformation with
T
[
3
−
2
]
=
[
6
2
]
T\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}6\\2\end{bmatrix}
T
[
3
−
2
]
=
[
6
2
]
and
T
[
−
5
4
]
=
[
−
4
0
]
T\begin{bmatrix}-5\\4\end{bmatrix}=\begin{bmatrix}-4\\0\end{bmatrix}
T
[
−
5
4
]
=
[
−
4
0
]
(a) Find the matrix of
T
T
T
, that is, find a matrix
A
A
A
such that
T
v
⃗
=
A
v
⃗
T\vec{v}=A\vec{v}
T
v
=
A
v
for all
v
⃗
∈
R
2
\vec{v}\in\mathbb{R}^2
v
∈
R
2
.
(b) Is
T
T
T
invertible? If so, find the matrix of
T
−
1
T^{-1}
T
−
1
. If not, prove why not.
The unit square in R2 is the set of all points (x; y) with 0 ≤ 𝑥 ≤ 2 and 0 ≤ 𝑦 ≤ 2. Suppose each point in
the unit square undergoes a linear transformation that has the given matrix representation
A
=
[
1
2
2
1
]
.Draw a labelled sketch of the transformed points.
A=\begin{bmatrix} 1&2\\2&1 \end{bmatrix}\text{.Draw a labelled sketch of the transformed points.}
A
=
[
1
2
2
1
]
.Draw a labelled sketch of the transformed points.
Find the matrix A representing the linear transformation T: R
3
to R
3
such that it satisfies:
𝑻(𝑒1) = 𝑒
1
+ 2𝑒
2
+ 3𝑒
3
, 𝑻(𝑒
1
+ 𝑒
2
) = 𝑒
1
+ 2𝑒
2
, 𝑻(𝑒
1
+ 𝑒
2
+ 𝑒
3
) = −2𝑒
2
+ 𝑒
3
.
Suppose T is a linear transformation such that
T
[
1
1
3
]
=
[
2
−
1
]
,
T
[
2
0
2
]
=
[
3
2
]
,
T
[
4
2
4
]
=
[
−
1
4
]
T\begin{bmatrix} 1\\1\\3 \end{bmatrix}=\begin{bmatrix} 2\\-1\\ \end{bmatrix},T\begin{bmatrix} 2\\0\\2 \end{bmatrix}=\begin{bmatrix} 3\\2 \end{bmatrix},T\begin{bmatrix} 4\\2\\4 \end{bmatrix}=\begin{bmatrix} -1\\4 \end{bmatrix}
T
1
1
3
=
[
2
−
1
]
,
T
2
0
2
=
[
3
2
]
,
T
4
2
4
=
[
−
1
4
]
Find the matrix of T.
Let
u
⃗
=
[
−
3
1
1
]
\vec{u} = \begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}
u
=
−
3
1
1
and let
T
:
R
3
→
R
3
T : \mathbb{R}^3\rightarrow\mathbb{R}^3
T
:
R
3
→
R
3
be the linear transformation defined by
T
(
v
⃗
)
=
proj
u
⃗
(
v
⃗
)
T(\vec{v})=\operatorname{proj}_{_{\vec{u}}}(\vec{v})
T
(
v
)
=
proj
u
(
v
)
. Find the matrix of
T
T
T
.
Find the matrix for the linear transformation S :R
3
to R
3
given by
S
(
[
x
y
z
]
)
=
(
[
−
y
+
z
x
+
y
+
z
−
x
+
2
y
]
)
S\left(\begin{bmatrix} x\\y\\z \end{bmatrix}\right)=\left(\begin{bmatrix} -y+z\\ x+y+z\\ -x+2y \end{bmatrix}\right)
S
x
y
z
=
−
y
+
z
x
+
y
+
z
−
x
+
2
y
Suppose
a
⃗
\vec{a}
a
is a fixed vector in ℝ
3
and consider the transformation 𝑇:ℝ
3
→ ℝ
3
defined by
𝑇
(
(
x
)
⃗
=
a
⃗
×
x
.
⃗
𝑇(\vec{\left(x\right)}=\vec{a}\ \times\vec{x.}
T
(
(
x
)
=
a
×
x
.
Prove that T is a linear transformation and find the matrix of T.
If T is a linear transformation such that
T
[
−
1
2
]
=
[
3
1
]
T\begin{bmatrix}-1\\2\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix}
T
[
−
1
2
]
=
[
3
1
]
and
T
[
1
3
]
=
[
−
1
2
]
T\begin{bmatrix}1\\3\end{bmatrix}=\begin{bmatrix}-1\\2\end{bmatrix}
T
[
1
3
]
=
[
−
1
2
]
, find
T
[
4
−
3
]
T\begin{bmatrix}4\\-3\end{bmatrix}
T
[
4
−
3
]
Suppose T is a linear transformation and
T
[
2
5
]
=
[
1
2
3
]
,
T
[
3
8
]
=
[
−
1
0
1
]
T\begin{bmatrix} 2\\5 \end{bmatrix}=\begin{bmatrix} 1\\2\\3 \end{bmatrix},T\begin{bmatrix} 3\\8 \end{bmatrix}=\begin{bmatrix} -1\\0\\1 \end{bmatrix}
T
[
2
5
]
=
1
2
3
,
T
[
3
8
]
=
−
1
0
1
Find the matrix of this linear transformation.
Consider the transformation T defined by
𝑇
(
𝑥
1
,
𝑥
2
,
𝑥
3
,
x
4
)
=
(
x
1
+
2
x
2
,
𝑥
3
−
𝑥
4
)
𝑇(𝑥_1,𝑥_2,𝑥_3,x_4)=(x_1+2x_2,𝑥_3−𝑥_4)
T
(
x
1
,
x
2
,
x
3
,
x
4
)
=
(
x
1
+
2
x
2
,
x
3
−
x
4
)
. Prove T is a linear transformation and find the matrix of T.
Suppose
a
⃗
\vec{a}
a
is a fixed vector in ℝ
n
and consider the transformation 𝑇:ℝ
n
→ ℝ defined by
𝑇
(
x
⃗
)
=
a
.
⃗
x
.
⃗
𝑇\ \left(\vec{x}\right)=\vec{a.}\ \ \vec{x.}\ \
T
(
x
)
=
a
.
x
.
Prove that T is a linear transformation and find the matrix of T.
Suppose T is a linear transformation and
T
[
2
5
]
=
[
1
2
3
]
,
T
[
3
8
]
=
[
−
1
0
1
]
T\begin{bmatrix} 2\\5 \end{bmatrix}=\begin{bmatrix} 1\\2\\3 \end{bmatrix},T\begin{bmatrix} 3\\8 \end{bmatrix}=\begin{bmatrix} -1\\0\\1 \end{bmatrix}
T
[
2
5
]
=
1
2
3
,
T
[
3
8
]
=
−
1
0
1
Find the matrix of this linear transformation.
For a linear transformation W from R
2
to R
2
, if 𝑊
W
(
[
2
0
]
)
=
[
3
2
]
,
W
(
[
1
2
]
)
=
[
−
1
4
]
,
f
i
n
d
W
(
[
−
4
4
]
)
.
W \left(\begin{bmatrix} 2\\0 \end{bmatrix}\right)=\begin{bmatrix} 3\\2 \end{bmatrix},W\left(\begin{bmatrix} 1\\2 \end{bmatrix}\right)=\begin{bmatrix} -1\\4 \end{bmatrix},find\ W\left(\begin{bmatrix} -4\\4 \end{bmatrix}\right).
W
(
[
2
0
]
)
=
[
3
2
]
,
W
(
[
1
2
]
)
=
[
−
1
4
]
,
f
in
d
W
(
[
−
4
4
]
)
.
True or False: Suppose
a
⃗
\vec{a\ }
a
is a fixed vector in
R
n
\mathbb{R}^n
R
n
and
T
:
R
n
→
R
T:\mathbb{R}^n\to\mathbb{R}
T
:
R
n
→
R
defined by
T
(
x
⃗
)
=
a
⃗
⋅
x
⃗
T(\vec{x})=\vec{a}\cdot\vec{x}
T
(
x
)
=
a
⋅
x
then
T
T
T
is a linear transformation and its matrix is
[
a
1
⋯
a
n
]
\begin{bmatrix}a_1&\cdots&a_n\end{bmatrix}
[
a
1
⋯
a
n
]
Suppose
T
:
R
2
→
R
3
T:\mathbb{R}^2\to\mathbb{R}^3
T
:
R
2
→
R
3
is a linear transformation and
T
[
2
5
]
=
[
1
2
3
]
,
T
[
3
8
]
=
[
−
1
0
1
]
T\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad T\begin{bmatrix}3\\8\end{bmatrix}=\begin{bmatrix}-1\\0\\1\end{bmatrix}
T
[
2
5
]
=
1
2
3
,
T
[
3
8
]
=
−
1
0
1
Find the matrix of this linear transformation.
Example: Finding the Linear Transformation Matrix
Find the corresponding matrix M of the linear transformation 𝑓(𝑥, 𝑦, 𝑧) = (𝑥 − 𝑦, 3𝑦, 3𝑥 + 2𝑧).
Finding Non-linear Transformation
Which of the following transformations are linear?
Is the given transformation linear?
Let
V
V
V
be the vector space
{
(
x
,
y
)
∣
x
,
y
∈
R
+
}
\{(x,y)\;|\;x,y\in\mathbb{R}^+\}
{(
x
,
y
)
∣
x
,
y
∈
R
+
}
with the operations:
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
(x_1,y_1)+(x_2,y_2)=(x_1x_2,y_1y_2)
(
x
1
,
y
1
)
+
(
x
2
,
y
2
)
=
(
x
1
x
2
,
y
1
y
2
)
k
⋅
(
x
,
y
)
=
(
x
k
,
y
k
)
k\cdot(x,y)=(x^k,y^k)
k
⋅
(
x
,
y
)
=
(
x
k
,
y
k
)
Practice Question: Linear and Non-Linear Maps
Determine whether the following maps are linear or not.
1.
L
:
R
→
R
L: \mathbb{R} \rightarrow \mathbb{R}
L
:
R
→
R
, defined by
L
(
x
)
=
cos
(
x
)
L(x) = \cos(x)
L
(
x
)
=
cos
(
x
)
.
2.
L
:
R
→
R
L : \mathbb{R} \rightarrow \mathbb{R}
L
:
R
→
R
, defined by
L
(
x
)
=
3
x
−
2
L(x) = 3x -2
L
(
x
)
=
3
x
−
2
19.4F_Mid_Builder_$\tkcth{8.4.}\tkcf{15}$_$\tkco{Mock 3}$.
The following is known about the system of linear equations given by
M
‾
x
⃗
=
y
⃗
\M \, \vx = \vec{y}
M
x
=
y
:
There is a unique solution for any
y
⃗
∈
R
5
\vec{y} \, \in \, \mathbb{R}^{5}
y
∈
R
5
.
If one interchanges the 3rd and 4th rows and then adds twice the third row to the first before finally multiplying the 2nd and 4th rows by 3,
M
‾
\M
M
will be in its RREF.
Find the matrix of a given transformation
(a)
Let
L
:
R
3
⟶
R
4
L:\mathbb{R}^3\longrightarrow\mathbb{R}^4
L
:
R
3
⟶
R
4
be defined by:
L
(
(
x
,
y
,
z
)
)
=
(
2
x
−
y
,
2
y
+
3
z
,
−
3
x
+
5
z
,
x
−
y
+
z
)
L\big((x,y,z)\big)=(2x-y,2y+3z,-3x+5z,x-y+z)
L
(
(
x
,
y
,
z
)
)
=
(
2
x
−
y
,
2
y
+
3
z
,
−
3
x
+
5
z
,
x
−
y
+
z
)
Find the matrix
A
A
A
that represents
L
L
L
19.4F_Mid_Builder_$\tkcth{8.4.}\tkcf{14}$_
Given the transformation
T
:
R
2
→
R
2
T: \mathbb{R}^2 \, \to \, \mathbb{R}^2
T
:
R
2
→
R
2
where
T
(
v
⃗
)
=
x
⃗
T(\vv) = \vx
T
(
v
)
=
x
and
T
(
w
⃗
)
=
y
⃗
T(\vw) = \vec{y}
T
(
w
)
=
y
, where
v
⃗
=
[
1
4
]
\vv = \colvec{1}{4}
v
=
[
1
4
]
,
w
⃗
=
[
3
11
]
\vw = \colvec{3}{11}
w
=
[
3
11
]
,
x
⃗
=
[
0
2
]
\vx = \colvec{0}{2}
x
=
[
0
2
]
and
y
⃗
=
[
−
3
0
]
\vec{y} = \colvec{-3}{0}
y
=
[
−
3
0
]
,
find the matrix representation
M
‾
T
B
\M_{T_B}
M
T
B
of the transformation
T
T
T
in the basis
B
=
{
[
2
8
]
,
[
−
3
−
11
]
}
B = \left\{ \colvec{2}{8},\, \colvec{-3}{-11} \right\}
B
=
{
[
2
8
]
,
[
−
3
−
11
]
}
.
Let
L
:
R
2
→
R
3
L : \mathbb{R}^2 \rightarrow \mathbb{R}^{3}
L
:
R
2
→
R
3
be the map defined by
L
(
x
,
y
)
=
(
2
x
,
3
y
,
x
+
y
)
L(x, y) = (2x, 3y, x + y)
L
(
x
,
y
)
=
(
2
x
,
3
y
,
x
+
y
)
Is this map a linear transformation? If so, what is the kernel and image of this function?
133 - FML 3 - 18.1W - e.g. 34
If
T
[
1
0
]
=
[
4
−
4
]
\bcb{T\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 4 \\ -4 \end{bmatrix}}
T
[
1
0
]
=
[
4
−
4
]
and
T
[
0
1
]
=
[
3
−
1
]
\bcb{T\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}}
T
[
0
1
]
=
[
3
−
1
]
, find the matrix representation of the transformation in the space spanned by the basis vectors
u
⃗
1
=
(
1
,
1
)
\bcb{\vec{u}_1 = (1,1)}
u
1
=
(
1
,
1
)
and
u
⃗
2
=
(
1
,
−
1
)
\bcb{\vec{u}_2 = (1,-1)}
u
2
=
(
1
,
−
1
)
.
M
‾
T
{
u
⃗
1
,
u
⃗
2
}
=
\M_{T_{\{\vu_1, \vu_2\}}} =
M
T
{
u
1
,
u
2
}
=
7
1
-5
-3
19.4F_Mid_Builder_$\tkcth{8.5.}\tkco{5}$__$\tkco{Mock 3}$
Given the basis for
R
2
\mathbb{R}^2
R
2
v
⃗
=
[
1
−
1
]
\vv = \colvec{1}{-1}
v
=
[
1
−
1
]
and
w
⃗
=
[
2
−
3
]
\vw = \colvec{2}{-3}
w
=
[
2
−
3
]
.
If
T
(
v
⃗
)
=
2
v
⃗
+
3
w
⃗
T(\vv) = 2\vv + 3\vw
T
(
v
)
=
2
v
+
3
w
and
T
(
w
⃗
)
=
v
⃗
+
3
w
⃗
T(\vw) = \vv + 3\vw
T
(
w
)
=
v
+
3
w
, determine the matrix representation of the transformation
T
T
T
.
Let
L
:
R
3
⟶
R
L:\mathbb{R}^3\longrightarrow\mathbb{R}
L
:
R
3
⟶
R
be defined by:
L
(
(
x
,
y
,
z
)
)
=
3
x
−
y
+
2
z
L\big((x,y,z)\big)=3x-y+2z
L
(
(
x
,
y
,
z
)
)
=
3
x
−
y
+
2
z
(a)
Show that
v
→
=
(
2
,
−
10
,
−
8
)
\overrightarrow{v}=(2,-10,-8)
v
=
(
2
,
−
10
,
−
8
)
is in
k
e
r
(
L
)
ker(L)
k
er
(
L
)
What is the matrix for the linear map
L
(
x
,
y
,
z
)
=
(
x
+
y
+
z
,
2
x
−
3
z
)
L(x, y, z) = (x + y + z, 2x - 3z)
L
(
x
,
y
,
z
)
=
(
x
+
y
+
z
,
2
x
−
3
z
)
with respect to the standard basis?
Let
L
L
L
be a linear transformation defined by:
L
(
x
→
)
=
A
x
→
L(\overrightarrow{x})=A\overrightarrow{x}
L
(
x
)
=
A
x
where
A
=
[
2
−
3
1
−
1
4
−
3
]
A=\left[\begin{array}{rrr} 2&-3&1\\ -1&4&-3 \end{array}\right]
A
=
[
2
−
1
−
3
4
1
−
3
]
Properties
Practice: Similar Matrices
Consider the matrix
A
=
[
−
1
4
2
1
]
A= \left[ \begin{array}{rrr} -1 & 4\\ 2 & 1 \end{array} \right]
A
=
[
−
1
2
4
1
]
.
Concept Clarifier
If
T
T
T
and
S
S
S
are linear transformations induced by the matrices
[
−
2
0
2
4
]
,
[
1
−
2
0
1
]
\begin{bmatrix}-2&0\\2&4\end{bmatrix},\quad \begin{bmatrix}1&-2\\0&1\end{bmatrix}
[
−
2
2
0
4
]
,
[
1
0
−
2
1
]
respectively, then find
(
S
∘
T
)
(
x
⃗
)
(S \circ T)(\vec{x})
(
S
∘
T
)
(
x
)
where
x
⃗
=
[
1
5
]
\vec{x}=\begin{bmatrix}1\\5\end{bmatrix}
x
=
[
1
5
]
133 - FML 3 - 18.1W e.g. 2.3
Determine whether the following transformation is linear:
{
y
1
=
3
(
x
1
+
x
2
)
y
2
=
−
x
1
\boldsymbol{ \begin{cases} y_1 &=~ 3(x_1 + x_2) \\ y_2 & = -x_1 \end{cases} }
{
y
1
y
2
=
3
(
x
1
+
x
2
)
=
−
x
1
133 - FML 3 - 18.1W e.g. 2.3
Determine whether the following transformation is linear:
{
y
1
=
1
y
2
=
1
y
3
=
1
\boldsymbol{ \begin{cases} y_1 &=~ 1 \\ y_2 & = 1 \\ y_3 &=~ 1 \end{cases} }
⎩
⎨
⎧
y
1
y
2
y
3
=
1
=
1
=
1
133 - FML 3 - 18.1W e.g. 2.2
Determine whether the following transformation is linear:
{
y
1
=
x
1
y
2
=
x
1
x
2
\boldsymbol{ \begin{cases} y_1 &=~ x_1 \\ y_2 & = x_1x_2 \end{cases} }
{
y
1
y
2
=
x
1
=
x
1
x
2
133 - FML 3 - 18.1W e.g. 2.1
Determine whether the following transformation is linear:
{
y
1
=
x
3
y
2
=
x
3
y
3
=
x
3
\boldsymbol{ \begin{cases} y_1 &=~ x_3 \\ y_2 & = ~x_3 \\ y_3 &=~ x_3 \end{cases} }
⎩
⎨
⎧
y
1
y
2
y
3
=
x
3
=
x
3
=
x
3
Linear Transformations
Practice: Linear Transformations
Suppose
T
:
R
2
→
R
3
T:\mathbb{R}^2\to\mathbb{R}^3
T
:
R
2
→
R
3
is a linear transformation where
T
[
2
5
]
=
[
1
2
3
]
,
T
[
3
8
]
=
[
−
1
0
1
]
T\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad T\begin{bmatrix}3\\8\end{bmatrix}=\begin{bmatrix}-1\\0\\1\end{bmatrix}
T
[
2
5
]
=
1
2
3
,
T
[
3
8
]
=
−
1
0
1
.
Let
L
:
R
3
⟶
R
2
L:\mathbb{R}^3\longrightarrow\mathbb{R}^2
L
:
R
3
⟶
R
2
be defined by:
L
(
(
x
,
y
,
z
)
)
=
(
x
+
2
y
+
3
z
,
−
x
+
2
y
+
5
z
)
L\big((x,y,z)\big)=(x+2y+3z,-x+2y+5z)
L
(
(
x
,
y
,
z
)
)
=
(
x
+
2
y
+
3
z
,
−
x
+
2
y
+
5
z
)
(a)
Prove that
L
L
L
is a linear transformation
Define the vector space
V
=
{
(
x
,
y
,
z
)
∣
x
,
y
,
z
>
0
}
V=\{(x,y,z)\;|\;x,y,z>0\}
V
=
{(
x
,
y
,
z
)
∣
x
,
y
,
z
>
0
}
with the operations:
(
x
1
,
y
1
,
z
1
)
+
(
x
2
,
y
2
,
z
2
)
=
(
x
1
x
2
,
y
1
y
2
,
z
1
+
z
2
)
(x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1x_2,y_1y_2,\textcolor{red}{z_1+z_2})
(
x
1
,
y
1
,
z
1
)
+
(
x
2
,
y
2
,
z
2
)
=
(
x
1
x
2
,
y
1
y
2
,
z
1
+
z
2
)
k
⋅
(
x
,
y
,
z
)
=
(
x
k
,
y
k
,
k
z
)
k\cdot(x,y,z)=(x^k,y^k,\textcolor{red}{kz})
k
⋅
(
x
,
y
,
z
)
=
(
x
k
,
y
k
,
k
z
)
Let
L
L
L
be the linear transformation such that
L
(
v
→
)
=
A
v
→
L(\overrightarrow{v})=A\overrightarrow{v}
L
(
v
)
=
A
v
where:
A
=
[
1
1
1
−
1
0
2
0
1
0
2
1
−
1
]
A=\left[\begin{array}{rrr} 1&1&1\\[0.5em] -1&0&2\\[0.5em] 0&1&0\\[0.5em] 2&1&-1\\ \end{array}\right]
A
=
1
−
1
0
2
1
0
1
1
1
2
0
−
1
Find the rule that defines
L
L
L
including the domain and co-domain vector spaces
Introduction to Transformations
Practice: Linear Transformations
Suppose
T
:
R
2
→
R
3
T:\mathbb{R}^2\to\mathbb{R}^3
T
:
R
2
→
R
3
is a linear transformation and
T
[
2
5
]
=
[
1
2
3
]
,
T
[
3
8
]
=
[
−
1
0
1
]
T\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix},\quad T\begin{bmatrix}3\\8\end{bmatrix}=\begin{bmatrix}-1\\0\\1\end{bmatrix}
T
[
2
5
]
=
1
2
3
,
T
[
3
8
]
=
−
1
0
1
Find the matrix of this linear transformation.
Linear Transformations
Given that
S
:
R
2
→
R
2
S:\mathbb{R}^2 \to \mathbb{R}^2
S
:
R
2
→
R
2
and
T
:
R
2
→
R
2
T:\mathbb{R}^2 \to \mathbb{R}^2
T
:
R
2
→
R
2
such that
S
[
3
5
]
=
[
1
−
1
]
,
S
[
3
−
1
]
=
[
2
5
]
S\begin{bmatrix}3\\5\end{bmatrix}=\begin{bmatrix}1\\-1\end{bmatrix}, \quad S\begin{bmatrix}3\\-1\end{bmatrix}=\begin{bmatrix}2\\5\end{bmatrix}
S
[
3
5
]
=
[
1
−
1
]
,
S
[
3
−
1
]
=
[
2
5
]
and
T
[
2
−
2
]
=
[
1
0
]
,
T
[
1
1
]
=
[
2
5
]
T\begin{bmatrix}2\\-2\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}, \quad T\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}2\\5\end{bmatrix}
T
[
2
−
2
]
=
[
1
0
]
,
T
[
1
1
]
=
[
2
5
]
Find
S
∘
T
[
3
−
1
]
S \circ T\begin{bmatrix}3\\-1\end{bmatrix}
S
∘
T
[
3
−
1
]
Matrix Multiplication
Find a 3 × 3 matrix M such that
M
‾
[
0
−
2
0
]
=
[
−
1
1
−
1
]
,
M
‾
[
0
0
1
8
]
=
[
0
−
2
3
]
,
and
M
‾
[
1
3
0
0
]
=
[
−
2
2
−
1
]
\bcb{ \M\begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 0 \end{bmatrix}= \begin{bmatrix} -1\\\ \ \ 1\\-1 \end{bmatrix}\!, \hspace{0.7cm} \M\begin{bmatrix} 0\\0\\\tfrac{1}{8} \end{bmatrix}= \begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 3 \end{bmatrix}\!, \hspace{0.5cm} \text{\small{and}}\hspace{0.5cm} \M\begin{bmatrix} \tfrac{1}{3}\\0\\0 \end{bmatrix}= \begin{bmatrix} -2\\\ \ \ 2\\-1 \end{bmatrix} }
M
0
−
2
0
=
−
1
1
−
1
,
M
0
0
8
1
=
0
−
2
3
,
and
M
3
1
0
0
=
−
2
2
−
1
Matrix of a Projection
a) Find the matrix
A
A
A
so that for
x
⃗
∈
R
3
\vec x \in \mathbb{R}^3
x
∈
R
3
the transformation mapping
x
⃗
→
p
r
o
j
(
−
1
3
,
1
,
1
2
)
x
⃗
\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec x
x
→
p
r
o
j
(
3
−
1
,
1
,
2
1
)
x
is the same as the transformation mapping
x
⃗
→
A
x
⃗
\vec x\rightarrow A\vec x
x
→
A
x
b) What is
r
a
n
k
(
A
)
rank(A)
r
ank
(
A
)
?
Matrix Multiplication
e.g. Find a 3 × 3 matrix M such that
M
[
0
−
2
0
]
=
[
−
1
1
−
1
]
,
M
[
0
0
0
]
=
[
0
−
2
3
]
,
and
M
[
1
3
0
0
]
=
[
−
2
2
−
1
]
\!M\begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 0 \end{bmatrix}= \begin{bmatrix} -1\\\ \ \ 1\\-1 \end{bmatrix}\!, \hspace{0.7cm} \!M\begin{bmatrix} 0\\0\\0 \end{bmatrix}= \begin{bmatrix} \ \ \ 0\\-2\\\ \ \ 3 \end{bmatrix}\!, \hspace{0.7cm} \text{\small{and}}\ M\begin{bmatrix} \frac{1}{3}\\0\\0 \end{bmatrix}= \begin{bmatrix} -2\\\ \ \ 2\\-1 \end{bmatrix}
M
0
−
2
0
=
−
1
1
−
1
,
M
0
0
0
=
0
−
2
3
,
and
M
3
1
0
0
=
−
2
2
−
1
Given the following matrix, what can you conclude?
A
=
[
1
0
−
2
1
5
1
]
A = \begin{bmatrix} 1 & 0 \\ -2 & 1\\ 5 & 1 \end{bmatrix}
A
=
1
−
2
5
0
1
1
Projection onto a subspace
Suppose
u
⃗
,
v
⃗
∈
R
n
\vec u,\vec v\in\mathbb{R}^n
u
,
v
∈
R
n
satisfy
u
⃗
⋅
v
⃗
=
0
,
∣
∣
u
⃗
∣
∣
=
∣
∣
v
⃗
∣
∣
=
1
\vec u\cdot\vec v=0, ||\vec u||=||\vec v||=1
u
⋅
v
=
0
,
∣∣
u
∣∣
=
∣∣
v
∣∣
=
1
, and define
U
:
=
s
p
a
n
{
u
⃗
,
v
⃗
}
U := span\{\vec u,\vec v\}
U
:=
s
p
an
{
u
,
v
}
.
Using the definition that
p
r
o
j
U
x
⃗
=
A
(
A
T
A
)
−
1
A
T
x
⃗
proj_U\vec x=A(A^TA)^{-1}A^T\vec x
p
r
o
j
U
x
=
A
(
A
T
A
)
−
1
A
T
x
, where
A
=
[
u
⃗
v
⃗
]
A=\left[\begin{array}{rr}\vec u&\vec v\end{array}\right]
A
=
[
u
v
]
, show that for all
x
⃗
∈
R
n
\vec x\in\mathbb{R}^n
x
∈
R
n
we have that
p
r
o
j
U
x
⃗
=
p
r
o
j
u
⃗
x
⃗
+
p
r
o
j
v
⃗
x
⃗
proj_U\vec x=proj_{\vec u}\vec x+proj_{\vec v}\vec x
p
r
o
j
U
x
=
p
r
o
j
u
x
+
p
r
o
j
v
x
.