Exercises
Determine the
transformation matrices
M
v
u
{\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}}
and
M
u
v
{\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}}
,
for the
standard basis
u
{\displaystyle {}{\mathfrak {u}}}
, and the basis
v
{\displaystyle {}{\mathfrak {v}}}
in
R
4
{\displaystyle {}\mathbb {R} ^{4}}
, which is given by
v
1
=
(
0
0
1
0
)
,
v
2
=
(
1
0
0
0
)
,
v
3
=
(
0
0
0
1
)
,
v
4
=
(
0
1
0
0
)
.
{\displaystyle v_{1}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}},\,v_{2}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}},\,v_{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}},\,v_{4}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}.}
Determine the
transformation matrices
M
v
u
{\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}}
and
M
u
v
{\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}}
,
for the
standard basis
u
{\displaystyle {}{\mathfrak {u}}}
, and the basis
v
{\displaystyle {}{\mathfrak {v}}}
of
R
3
{\displaystyle {}\mathbb {R} ^{3}}
, which is given by the vectors
v
1
=
(
4
5
1
)
,
v
2
=
(
2
3
−
8
)
and
v
3
=
(
5
7
−
3
)
{\displaystyle v_{1}={\begin{pmatrix}4\\5\\1\end{pmatrix}},\,\,v_{2}={\begin{pmatrix}2\\3\\-8\end{pmatrix}}\,{\text{ and }}\,v_{3}={\begin{pmatrix}5\\7\\-3\end{pmatrix}}}
Let
v
=
v
1
,
v
2
,
v
3
{\displaystyle {}{\mathfrak {v}}=v_{1},v_{2},v_{3}}
be a
basis
of a three-dimensional
K
{\displaystyle {}K}
-vector space
V
{\displaystyle {}V}
.
a) Show that
w
=
v
1
,
v
1
+
v
2
,
v
2
+
v
3
{\displaystyle {}{\mathfrak {w}}=v_{1},v_{1}+v_{2},v_{2}+v_{3}}
is also a basis of
V
{\displaystyle {}V}
.
b) Determine the
transformation matrix
M
v
w
{\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {w}}}
.
c) Determine the transformation matrix
M
w
v
{\displaystyle {}M_{\mathfrak {w}}^{\mathfrak {v}}}
.
d) Compute the coordinates with respect to the basis
v
{\displaystyle {}{\mathfrak {v}}}
for the vector, which has the coordinates
(
4
8
−
9
)
{\displaystyle {}{\begin{pmatrix}4\\8\\-9\end{pmatrix}}}
with respect to the basis
w
{\displaystyle {}{\mathfrak {w}}}
.
e) Compute the coordinates with respect to the basis
w
{\displaystyle {}{\mathfrak {w}}}
for the vector, which has the coordinates
(
3
−
7
5
)
{\displaystyle {}{\begin{pmatrix}3\\-7\\5\end{pmatrix}}}
with respect to the basis
v
{\displaystyle {}{\mathfrak {v}}}
.
Determine the
transformation matrices
M
v
u
{\displaystyle {}M_{\mathfrak {v}}^{\mathfrak {u}}}
and
M
u
v
{\displaystyle {}M_{\mathfrak {u}}^{\mathfrak {v}}}
,
for the
standard basis
u
{\displaystyle {}{\mathfrak {u}}}
, and the basis
v
{\displaystyle {}{\mathfrak {v}}}
of
C
2
{\displaystyle {}\mathbb {C} ^{2}}
, which is given by the vectors
v
1
=
(
3
+
5
i
1
−
i
)
and
v
2
=
(
2
+
3
i
4
+
i
)
.
{\displaystyle v_{1}={\begin{pmatrix}3+5{\mathrm {i} }\\1-{\mathrm {i} }\end{pmatrix}}{\text{ and }}v_{2}={\begin{pmatrix}2+3{\mathrm {i} }\\4+{\mathrm {i} }\end{pmatrix}}.}
We consider the families of vectors
v
=
(
7
−
4
)
,
(
8
1
)
and
u
=
(
4
6
)
,
(
7
3
)
{\displaystyle {\mathfrak {v}}={\begin{pmatrix}7\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\1\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}4\\6\end{pmatrix}},\,{\begin{pmatrix}7\\3\end{pmatrix}}}
in
R
2
{\displaystyle {}\mathbb {R} ^{2}}
.
a) Show that
v
{\displaystyle {}{\mathfrak {v}}}
and
u
{\displaystyle {}{\mathfrak {u}}}
are both a
basis
of
R
2
{\displaystyle {}\mathbb {R} ^{2}}
.
b) Let
P
∈
R
2
{\displaystyle {}P\in \mathbb {R} ^{2}}
denote the point which has the coordinates
(
−
2
,
5
)
{\displaystyle {}(-2,5)}
with respect to the basis
v
{\displaystyle {}{\mathfrak {v}}}
. What are the coordinates of this point with respect to the basis
u
{\displaystyle {}{\mathfrak {u}}}
?
c) Determine the
transformation matrix
which describes the
change of bases
from
v
{\displaystyle {}{\mathfrak {v}}}
to
u
{\displaystyle {}{\mathfrak {u}}}
.
We consider the linear map
φ
:
K
3
⟶
K
2
,
(
x
y
z
)
⟼
(
1
2
5
4
1
1
)
(
x
y
z
)
.
{\displaystyle \varphi \colon K^{3}\longrightarrow K^{2},{\begin{pmatrix}x\\y\\z\end{pmatrix}}\longmapsto {\begin{pmatrix}1&2&5\\4&1&1\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}.}
Let
U
⊆
K
3
{\displaystyle {}U\subseteq K^{3}}
be the subspace of
K
3
{\displaystyle {}K^{3}}
, defined by the linear equation
2
x
+
3
y
+
4
z
=
0
{\displaystyle {}2x+3y+4z=0}
,
and let
ψ
{\displaystyle {}\psi }
be the restriction of
φ
{\displaystyle {}\varphi }
on
U
{\displaystyle {}U}
. On
U
{\displaystyle {}U}
, there are given vectors of the form
u
=
(
0
,
1
,
a
)
,
v
=
(
1
,
0
,
b
)
and
w
=
(
1
,
c
,
0
)
.
{\displaystyle u=(0,1,a),\,v=(1,0,b){\text{ and }}w=(1,c,0).}
Compute the "change of basis" matrix between the bases
b
1
=
v
,
w
,
b
2
=
u
,
w
and
b
3
=
u
,
v
{\displaystyle {\mathfrak {b}}_{1}=v,w,\,{\mathfrak {b}}_{2}=u,w{\text{ and }}{\mathfrak {b}}_{3}=u,v}
of
U
{\displaystyle {}U}
, and the transformation matrix of
ψ
{\displaystyle {}\psi }
with respect to these three bases
(and the standard basis of
K
2
{\displaystyle {}K^{2}}
).
Let
K
{\displaystyle {}K}
be a
field ,
and let
V
{\displaystyle {}V}
and
W
{\displaystyle {}W}
be
K
{\displaystyle {}K}
-vector spaces .
Let
φ
:
V
⟶
W
{\displaystyle \varphi \colon V\longrightarrow W}
be a linear map. Prove that for all vectors
v
1
,
…
,
v
n
∈
V
{\displaystyle {}v_{1},\ldots ,v_{n}\in V}
and coefficients
s
1
,
…
,
s
n
∈
K
{\displaystyle {}s_{1},\ldots ,s_{n}\in K}
,
the relationship
φ
(
∑
i
=
1
n
s
i
v
i
)
=
∑
i
=
1
n
λ
i
φ
(
v
i
)
{\displaystyle {}\varphi {\left(\sum _{i=1}^{n}s_{i}v_{i}\right)}=\sum _{i=1}^{n}\lambda _{i}\varphi (v_{i})\,}
holds.
Let
K
{\displaystyle {}K}
be a
field ,
and let
V
{\displaystyle {}V}
be a
K
{\displaystyle {}K}
-vector space .
Prove that for
a
∈
K
{\displaystyle {}a\in K}
the map
V
⟶
V
,
v
⟼
a
v
,
{\displaystyle V\longrightarrow V,v\longmapsto av,}
is linear.[ 1]
Interpret the following physical laws as linear functions from
R
{\displaystyle {}\mathbb {R} }
to
R
{\displaystyle {}\mathbb {R} }
. Establish, in each situation, what is the measurable variable and what is the proportionality factor.
Mass is volume times density.
Energy is mass times the calorific value.
The distance is speed multiplied by time.
Force is mass times acceleration.
Energy is force times distance.
Energy is power times time.
Voltage is resistance times electric current.
Charge is current multiplied by time.
Around the Earth along the equator, a ribbon is placed. However, the ribbon is one meter longer than the equator, so that it is lifted uniformly all around to be tense. Which of the following creatures can run/fly/swim/dance under it?
An amoeba.
An ant.
A tit.
A flounder.
A boa constrictor.
A guinea pig.
A boa constrictor that has swallowed a guinea pig.
A very good limbo dancer.
Suppose that a linear function
φ
:
Q
⟶
Q
{\displaystyle \varphi \colon \mathbb {Q} \longrightarrow \mathbb {Q} }
has for
11
13
{\displaystyle {}{\frac {11}{13}}}
the value
7
17
{\displaystyle {}{\frac {7}{17}}}
. What is its vale for
3
19
{\displaystyle {}{\frac {3}{19}}}
?
Which of the following functions
f
:
R
→
R
{\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }
are
linear ?
The
real exponential function .
The zero function.
The constant function with value
7
{\displaystyle {}7}
.
The squaring function
x
↦
x
2
{\displaystyle {}x\mapsto x^{2}}
.
The function which halves every real number.
The function which subtracts
1
{\displaystyle {}1}
from every real number.
Which of the following geometric shapes can be the image of a square under a
linear mapping
from
R
2
{\displaystyle {}\mathbb {R} ^{2}}
to
R
2
{\displaystyle {}\mathbb {R} ^{2}}
?
Consider the linear map
φ
:
R
2
⟶
R
{\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} }
such that
φ
(
1
3
)
=
5
and
φ
(
2
−
3
)
=
4.
{\displaystyle \varphi {\begin{pmatrix}1\\3\end{pmatrix}}=5{\text{ and }}\varphi {\begin{pmatrix}2\\-3\end{pmatrix}}=4.}
Compute
φ
(
7
6
)
.
{\displaystyle \varphi {\begin{pmatrix}7\\6\end{pmatrix}}.}
Complete the proof of
the theorem on determination on basis ,
by proving the compatibility with the scalar multiplication.
Lucy Sonnenschein works as a bicycle messenger, she earns
12
{\displaystyle {}12}
€ per hour. At the fruit market, the price
(per
100
{\displaystyle {}100}
gram)
for raspberries is
3
{\displaystyle {}3}
€, for strawberries the price is
2
{\displaystyle {}2}
€, and for apples the price is
0
,
4
{\displaystyle {}0,4}
€. Describe the mapping, which assigns to any purchase of fruits, the time, how long Lucy has to work for it, as a composition of linear mappings.
Let
K
{\displaystyle {}K}
be a
field ,
and let
U
,
V
,
W
{\displaystyle {}U,V,W}
be
vector spaces
over
K
{\displaystyle {}K}
. Let
φ
:
U
→
V
{\displaystyle {}\varphi \colon U\rightarrow V}
and
ψ
:
V
→
W
{\displaystyle {}\psi \colon V\rightarrow W}
be
linear maps .
Prove that also the
composite mapping
ψ
∘
φ
:
U
⟶
W
{\displaystyle \psi \circ \varphi \colon U\longrightarrow W}
is a linear map.
Let
K
{\displaystyle {}K}
be a field, and let
V
{\displaystyle {}V}
and
W
{\displaystyle {}W}
be
K
{\displaystyle {}K}
-vector spaces. Let
φ
:
V
⟶
W
{\displaystyle \varphi \colon V\longrightarrow W}
be a bijective linear map. Prove that also the inverse map
φ
−
1
:
W
⟶
V
{\displaystyle \varphi ^{-1}\colon W\longrightarrow V}
is linear.
Let
K
{\displaystyle {}K}
be a field, and let
V
{\displaystyle {}V}
be a
K
{\displaystyle {}K}
-vector space. Let
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
be a family of vectors in
V
{\displaystyle {}V}
. Consider the map
φ
:
K
n
⟶
V
,
(
s
1
,
…
,
s
n
)
⟼
∑
i
=
1
n
s
i
v
i
,
{\displaystyle \varphi \colon K^{n}\longrightarrow V,(s_{1},\ldots ,s_{n})\longmapsto \sum _{i=1}^{n}s_{i}v_{i},}
and prove the following statements.
φ
{\displaystyle {}\varphi }
is injective if and only if
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
are linearly independent.
φ
{\displaystyle {}\varphi }
is surjective if and only if
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
is a system of generators for
V
{\displaystyle {}V}
.
φ
{\displaystyle {}\varphi }
is bijective if and only if
v
1
,
…
,
v
n
{\displaystyle {}v_{1},\ldots ,v_{n}}
form a basis.
Prove that the functions
C
⟶
R
,
z
⟼
Re
(
z
)
,
{\displaystyle \mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Re} \,{\left(z\right)},}
and
C
⟶
R
,
z
⟼
Im
(
z
)
,
{\displaystyle \mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \operatorname {Im} \,{\left(z\right)},}
are
R
{\displaystyle {}\mathbb {R} }
-linear maps. Prove that also the complex conjugation is
R
{\displaystyle {}\mathbb {R} }
-linear, but not
C
{\displaystyle {}\mathbb {C} }
-linear. Is the absolute value
C
⟶
R
,
z
⟼
|
z
|
,
{\displaystyle \mathbb {C} \longrightarrow \mathbb {R} ,z\longmapsto \vert {z}\vert ,}
R
{\displaystyle {}\mathbb {R} }
-linear?
Consider the function
f
:
R
⟶
R
,
{\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,}
which sends a rational number
q
∈
Q
{\displaystyle {}q\in \mathbb {Q} }
to
q
{\displaystyle {}q}
, and all the irrational numbers to
0
{\displaystyle {}0}
. Is this a linear map? Is it compatible with multiplication by a scalar?
Let
K
{\displaystyle {}K}
be a field, and let
V
{\displaystyle {}V}
and
W
{\displaystyle {}W}
be
K
{\displaystyle {}K}
-vector spaces. Let
φ
:
V
⟶
W
{\displaystyle \varphi \colon V\longrightarrow W}
be a linear map. Prove the following facts.
For a linear subspace
S
⊆
V
{\displaystyle {}S\subseteq V}
,
also the image
φ
(
S
)
{\displaystyle {}\varphi (S)}
is a linear subspace of
W
{\displaystyle {}W}
.
In particular, the image
Im
φ
=
φ
(
V
)
{\displaystyle {}\operatorname {Im} \varphi =\varphi (V)\,}
of the map is a subspace of
W
{\displaystyle {}W}
.
For a linear subspace
T
⊆
W
{\displaystyle {}T\subseteq W}
,
also the preimage
φ
−
1
(
T
)
{\displaystyle {}\varphi ^{-1}(T)}
is a linear subspace of
V
{\displaystyle {}V}
.
In particular,
φ
−
1
(
0
)
{\displaystyle {}\varphi ^{-1}(0)}
is a subspace of
V
{\displaystyle {}V}
.
Find, by elementary geometric considerations, a matrix describing a rotation by 45 degrees counter-clockwise in the plane.
Prove the addition theorems for sine and cosine, using the rotation matrices.
Determine the kernel of the linear map
R
4
⟶
R
3
,
(
x
y
z
w
)
⟼
(
2
1
5
2
3
−
2
7
−
1
2
−
1
−
4
3
)
(
x
y
z
w
)
.
{\displaystyle \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{3},{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}\longmapsto {\begin{pmatrix}2&1&5&2\\3&-2&7&-1\\2&-1&-4&3\end{pmatrix}}{\begin{pmatrix}x\\y\\z\\w\end{pmatrix}}.}
Determine the kernel of the linear map
φ
:
R
4
⟶
R
2
,
{\displaystyle \varphi \colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{2},}
given by the matrix
M
=
(
2
3
0
−
1
4
2
2
5
)
.
{\displaystyle {}M={\begin{pmatrix}2&3&0&-1\\4&2&2&5\end{pmatrix}}\,.}
How does the
graph
of a
linear mapping
f
:
R
⟶
R
,
{\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,}
g
:
R
⟶
R
2
,
{\displaystyle g\colon \mathbb {R} \longrightarrow \mathbb {R} ^{2},}
h
:
R
2
⟶
R
{\displaystyle h\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} }
look like? How can you see in a sketch of the graph the kernel of the map?
Let
M
{\displaystyle {}M}
be an
m
×
n
{\displaystyle {}m\times n}
-matrix
over the field
K
{\displaystyle {}K}
, let
φ
:
K
n
→
K
m
{\displaystyle {}\varphi \colon K^{n}\rightarrow K^{m}}
be the corresponding
linear mapping ,
and let
M
x
=
c
{\displaystyle {}Mx=c}
denote
(depending on a vector
c
∈
K
m
{\displaystyle {}c\in K^{m}}
)
the corresponding
system of linear equations .
Show that the solution set of the system equals the
preimage
of
c
{\displaystyle {}c}
under the linear mapping
φ
{\displaystyle {}\varphi }
.
Let
V
{\displaystyle {}V}
and
W
{\displaystyle {}W}
be
vector spaces
over a
field
K
{\displaystyle {}K}
, and let
φ
,
ψ
:
V
→
W
{\displaystyle {}\varphi ,\psi \colon V\rightarrow W}
be
linear mappings .
Show that the mapping, defined by
(
φ
+
ψ
)
(
v
)
:=
φ
(
v
)
+
ψ
(
v
)
,
{\displaystyle {}(\varphi +\psi )(v):=\varphi (v)+\psi (v)\,,}
is also linear.
Give an example for a
linear mapping
φ
:
R
2
⟶
R
2
,
{\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},}
which is not
injective ,
but such that its
restriction
Q
2
⟶
R
2
{\displaystyle \mathbb {Q} ^{2}\longrightarrow \mathbb {R} ^{2}}
is injective.
We consider the mapping
Ψ
:
R
≥
0
4
⟶
R
≥
0
4
,
{\displaystyle \Psi \colon \mathbb {R} _{\geq 0}^{4}\longrightarrow \mathbb {R} _{\geq 0}^{4},}
which assigns to a four-tuple
(
a
,
b
,
c
,
d
)
{\displaystyle {}(a,b,c,d)}
the four-tuple
(
|
b
−
a
|
,
|
c
−
b
|
,
|
d
−
c
|
,
|
a
−
d
|
)
.
{\displaystyle (\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).}
Describe this mapping with a matrix, under the condition
a
≤
b
≤
c
≤
d
.
{\displaystyle {}a\leq b\leq c\leq d\,.}
Hand-in-exercises
Consider the linear map
φ
:
R
3
⟶
R
2
{\displaystyle \varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2}}
such that
φ
(
2
1
3
)
=
(
4
7
)
,
φ
(
0
4
2
)
=
(
1
1
)
and
φ
(
3
1
1
)
=
(
5
0
)
.
{\displaystyle \varphi {\begin{pmatrix}2\\1\\3\end{pmatrix}}={\begin{pmatrix}4\\7\end{pmatrix}},\,\varphi {\begin{pmatrix}0\\4\\2\end{pmatrix}}={\begin{pmatrix}1\\1\end{pmatrix}}{\text{ and }}\varphi {\begin{pmatrix}3\\1\\1\end{pmatrix}}={\begin{pmatrix}5\\0\end{pmatrix}}.}
Compute
φ
(
4
5
6
)
.
{\displaystyle \varphi {\begin{pmatrix}4\\5\\6\end{pmatrix}}.}
We consider the families of vectors
v
=
(
1
2
3
)
,
(
4
7
1
)
,
(
0
2
5
)
and
u
=
(
0
2
4
)
,
(
6
6
1
)
,
(
3
5
−
2
)
{\displaystyle {\mathfrak {v}}={\begin{pmatrix}1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}4\\7\\1\end{pmatrix}},\,{\begin{pmatrix}0\\2\\5\end{pmatrix}}\,\,{\text{ and }}\,\,{\mathfrak {u}}={\begin{pmatrix}0\\2\\4\end{pmatrix}},\,{\begin{pmatrix}6\\6\\1\end{pmatrix}},\,{\begin{pmatrix}3\\5\\-2\end{pmatrix}}}
in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
.
a) Show that
v
{\displaystyle {}{\mathfrak {v}}}
and
u
{\displaystyle {}{\mathfrak {u}}}
are both a
basis
of
R
3
{\displaystyle {}\mathbb {R} ^{3}}
.
b) Let
P
∈
R
3
{\displaystyle {}P\in \mathbb {R} ^{3}}
denote the point which has the coordinates
(
2
,
5
,
4
)
{\displaystyle {}(2,5,4)}
with respect to the basis
v
{\displaystyle {}{\mathfrak {v}}}
. What are the coordinates of this point with respect to the basis
u
{\displaystyle {}{\mathfrak {u}}}
?
c) Determine the
transformation matrix
which describes the
change of basis
from
v
{\displaystyle {}{\mathfrak {v}}}
to
u
{\displaystyle {}{\mathfrak {u}}}
.
Sketch the
image
of the pictured circles under the
linear mapping
given by the
matrix
(
2
0
0
3
)
{\displaystyle {}{\begin{pmatrix}2&0\\0&3\end{pmatrix}}}
from
R
2
{\displaystyle {}\mathbb {R} ^{2}}
to itself.
Find, by elementary geometric considerations, a matrix describing a rotation by 30 degrees counter-clockwise in the plane.
Determine the image and the kernel of the linear map
f
:
R
4
⟶
R
4
,
(
x
1
x
2
x
3
x
4
)
⟼
(
1
3
4
−
1
2
5
7
−
1
−
1
2
3
−
2
−
2
0
0
−
2
)
⋅
(
x
1
x
2
x
3
x
4
)
.
{\displaystyle f\colon \mathbb {R} ^{4}\longrightarrow \mathbb {R} ^{4},{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}1&3&4&-1\\2&5&7&-1\\-1&2&3&-2\\-2&0&0&-2\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}.}
Let
E
⊆
R
3
{\displaystyle {}E\subseteq \mathbb {R} ^{3}}
be the plane defined by the linear equation
5
x
+
7
y
−
4
z
=
0
{\displaystyle {}5x+7y-4z=0}
.
Determine a linear map
φ
:
R
2
⟶
R
3
,
{\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{3},}
such that the image of
φ
{\displaystyle {}\varphi }
is equal to
E
{\displaystyle {}E}
.
On the real vector space
G
=
R
4
{\displaystyle {}G=\mathbb {R} ^{4}}
of mulled wines, we consider the two linear maps
π
:
G
⟶
R
,
(
z
n
r
s
)
⟼
8
z
+
9
n
+
5
r
+
s
,
{\displaystyle \pi \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 8z+9n+5r+s,}
and
κ
:
G
⟶
R
,
(
z
n
r
s
)
⟼
2
z
+
n
+
4
r
+
8
s
.
{\displaystyle \kappa \colon G\longrightarrow \mathbb {R} ,{\begin{pmatrix}z\\n\\r\\s\end{pmatrix}}\longmapsto 2z+n+4r+8s.}
We consider
π
{\displaystyle {}\pi }
as the price function, and
κ
{\displaystyle {}\kappa }
as the caloric function. Determine a basis for
ker
(
π
)
{\displaystyle {}\operatorname {ker} {\left(\pi \right)}}
, one for
ker
(
κ
)
{\displaystyle {}\operatorname {ker} {\left(\kappa \right)}}
and one for
ker
(
π
×
κ
)
{\displaystyle {}\operatorname {ker} {\left(\pi \times \kappa \right)}}
.[ 2]
Footnotes
↑ Such a mapping is called a homothety , or a dilation with scale factor
a
{\displaystyle {}a}
.
↑ Do not mind that there may exist negative numbers. In a mulled wine, of course the ingredients do not enter with a negative coefficient. But if you would like to consider, for example, in how many ways you can change a particular recipe, without changing the total price or the total amount of energy, then the negative entries make sense.