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Consider the matrix transformations S and T induced by the matrices A=-1213 an…
Related Topics
Wize University Linear Algebra Textbook > Linear Transformations
Composition and Inverse
3 Activities
Consider the matrix transformations
S
S
S
and
T
T
T
induced by the matrices
A
=
[
−
1
2
1
3
]
A=\begin{bmatrix}-1&2\\1&3 \end{bmatrix}
A
=
[
−
1
1
2
3
]
and
B
=
[
1
2
2
−
1
]
B= \begin{bmatrix} 1&2\\2&-1 \end{bmatrix}
B
=
[
1
2
2
−
1
]
, respectively. Then
T
∘
S
T\circ S
T
∘
S
is;
Not a matrix transformation.
A matrix transformation induced by the matrix
[
0
4
3
2
]
\begin{bmatrix} 0&4\\3&2 \end{bmatrix}
[
0
3
4
2
]
A matrix transformation induced by the matrix
[
3
−
4
7
−
1
]
\begin{bmatrix} 3&-4\\7&-1 \end{bmatrix}
[
3
7
−
4
−
1
]
A matrix transformation induced by the matrix
[
1
8
−
3
1
]
\begin{bmatrix} 1&8\\-3&1 \end{bmatrix}
[
1
−
3
8
1
]
None of the above.
I don't know
Check Submission
More Composition and Inverse Questions:
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
Composition and Inverse
Practice: Composition of Linear Transformations
Let
T
:
R
2
→
R
3
T:\mathbb{R}^2\to\mathbb{R}^3
T
:
R
2
→
R
3
be a linear transformation defined by
T
[
x
y
]
=
[
x
+
y
3
y
x
+
2
y
]
T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x+2y\end{bmatrix}
T
[
x
y
]
=
x
+
y
3
y
x
+
2
y
.
Let
S
:
R
2
→
R
2
S:\reals^2 \to \reals^2
S
:
R
2
→
R
2
be the linear transformation induced by the matrix
B
=
[
1
1
0
1
]
B= \left[ \begin{array}{rrr} 1 & 1 \\ 0 & 1 \\ \end{array} \right]
B
=
[
1
0
1
1
]
.
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.
Choose all the matrix products that are the
identity matrix.
a) If A is a rotation matrix for 30 degree CCW, find 𝑛 such that: 𝑨
N
= A
-1
b) Find two different rotation matrices B such that
𝑩
𝟐
=
[
0
1
−
1
0
]
𝑩^𝟐=\begin{bmatrix} 0&1\\-1&0 \end{bmatrix}
B
2
=
[
0
−
1
1
0
]
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.
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
]
Composition and Inverse
Example: Composition of Linear Transformations
Suppose
T
T
T
and
S
S
S
are linear transformations induced by the matrices
A
=
[
−
2
0
2
4
]
,
B
=
[
1
0
−
2
1
]
A= \begin{bmatrix} -2&0\\ 2&4 \end{bmatrix}, \ B= \begin{bmatrix} 1&0\\ -2&1 \end{bmatrix}
A
=
[
−
2
2
0
4
]
,
B
=
[
1
−
2
0
1
]
, respectively.
Part 1)
Given the linear transformation
T
:
R
2
→
R
3
T:\mathbb{R}^2\to\mathbb{R}^3
T
:
R
2
→
R
3
defined by
T
[
x
y
]
=
[
x
+
y
3
y
x
−
2
y
]
T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y\\3y\\x-2y\end{bmatrix}
T
[
x
y
]
=
x
+
y
3
y
x
−
2
y
, find the vector
[
x
y
]
\begin{bmatrix}x\\y\end{bmatrix}
[
x
y
]
such that
T
[
x
y
]
=
[
2
1
1
]
T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\1\\1\end{bmatrix}
T
[
x
y
]
=
2
1
1
Composition of linear transformations
Suppose that
A
=
[
2
0
1
1
]
A=\left[\begin{array}{rr}2&0\\1&1\end{array}\right]
A
=
[
2
1
0
1
]
T
T
T
is the linear transformation induced by the matrix
A
A
A
, and
S
S
S
is the linear transformation with matrix
B
B
B
that rotates a vector by 60 degrees (
π
/
3
\pi/3
π
/3
rad) counterclockwise around the point
(
0
,
0
)
(0,0)
(
0
,
0
)
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
)
Let
{
e
1
,
…
,
e
n
}
\{ e_1, \dots, e_n \}
{
e
1
,
…
,
e
n
}
be a basis for an
n
n
n
- dimensional vector space
V
V
V
, and let
L
:
V
→
W
L : V\rightarrow W
L
:
V
→
W
be a linear map, where
W
W
W
is also a vector space. What conditions are necessary for
{
L
(
e
1
)
,
…
,
L
(
e
n
)
}
\{ L(e_1), \dots, L(e_n) \}
{
L
(
e
1
)
,
…
,
L
(
e
n
)}
to be a basis for
W
W
W
?
Practice: Composition and Matrix Inverse.
Suppose that
T
:
R
n
→
R
n
T:\mathbb{R}^n\rightarrow\mathbb{R}^n
T
:
R
n
→
R
n
is a linear transformation with inverse transformation
S
:
R
n
→
R
n
S:\mathbb{R}^n\rightarrow\mathbb{R}^n
S
:
R
n
→
R
n
, then, if
A
A
A
is the matrix of
T
T
T
, show that
A
A
A
is invertible and
A
−
1
A^{-1}
A
−
1
is the matrix of
S
S
S
.
Given that
T
:
R
2
→
R
2
T:\mathbb{R}^2 \to \mathbb{R}^2
T
:
R
2
→
R
2
is a linear transformation where:
T
[
3
0
]
=
[
6
−
9
]
T\begin{bmatrix}3\\0\end{bmatrix}=\begin{bmatrix}6\\-9\end{bmatrix}
T
[
3
0
]
=
[
6
−
9
]
and
T
[
0
1
]
=
[
1
1
]
T\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}1\\1\end{bmatrix}
T
[
0
1
]
=
[
1
1
]
Find
T
−
1
[
2
−
1
]
T^{-1}\begin{bmatrix}2\\-1\end{bmatrix}
T
−
1
[
2
−
1
]
Composition of Transformations
Inverse of Transformations