M
,
P
,
S
{\displaystyle {}M,P,S}
T
{\displaystyle {}T}
M
{\displaystyle {}M}
S
{\displaystyle {}S}
T
{\displaystyle {}T}
M
{\displaystyle {}M}
P
{\displaystyle {}P}
S
{\displaystyle {}S}
T
{\displaystyle {}T}
S
{\displaystyle {}S}
T
{\displaystyle {}T}
M
{\displaystyle {}M}
P
{\displaystyle {}P}
P
{\displaystyle {}P}
T
{\displaystyle {}T}
83
{\displaystyle {}83}
a) Set up a linear system of equations that expresses the conditions described.
We look at a clock with hour and minute hands. Now it is 6 o'clock, so that both hands have opposite directions. When will the hands have opposite directions again?
Find a
polynomial
f
=
a
+
b
X
+
c
X
2
{\displaystyle {}f=a+bX+cX^{2}\,}
with
a
,
b
,
c
∈
R
{\displaystyle {}a,b,c\in \mathbb {R} }
f
(
−
1
)
=
2
,
f
(
1
)
=
0
,
f
(
3
)
=
5.
{\displaystyle f(-1)=2,\,f(1)=0,\,f(3)=5.}
Find a
polynomial
f
=
a
+
b
X
+
c
X
2
+
d
X
3
{\displaystyle {}f=a+bX+cX^{2}+dX^{3}\,}
with
a
,
b
,
c
,
d
∈
R
{\displaystyle {}a,b,c,d\in \mathbb {R} }
f
(
0
)
=
1
,
f
(
1
)
=
2
,
f
(
2
)
=
0
,
f
(
−
1
)
=
1.
{\displaystyle f(0)=1,\,f(1)=2,\,f(2)=0,\,f(-1)=1.}
Exhibit a linear equation for the straight line in
R
2
{\displaystyle {}\mathbb {R} ^{2}}
(
2
,
3
)
{\displaystyle {}(2,3)}
(
5
,
−
7
)
{\displaystyle {}(5,-7)}
Determine an equation for the secant of the function
R
⟶
R
,
x
⟼
−
x
3
+
x
2
+
2
,
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto -x^{3}+x^{2}+2,}
to the points
3
{\displaystyle {}3}
4
{\displaystyle {}4}
Determine a linear equation for the plane in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
(
1
,
0
,
0
)
,
(
0
,
1
,
2
)
,
and
(
2
,
3
,
4
)
{\displaystyle (1,0,0),\,(0,1,2),{\text{ and }}(2,3,4)}
lie.
Given a
complex number
z
=
a
+
b
i
≠
0
,
{\displaystyle {}z=a+b{\mathrm {i} }\neq 0\,,}
find its inverse complex number with the help of a real system of linear equations, with two equations in two variables.
Solve, over the
complex numbers ,
the
linear system
of equations
i
x
+
y
+
(
2
−
i
)
z
=
2
7
y
+
2
i
z
=
−
1
+
3
i
(
2
−
5
i
)
z
=
1
.
{\displaystyle {\begin{matrix}{\mathrm {i} }x&+y&+(2-{\mathrm {i} })z&=&2\\&7y&+2{\mathrm {i} }z&=&-1+3{\mathrm {i} }\\&&(2-5{\mathrm {i} })z&=&1\,.\end{matrix}}}
Let
K
{\displaystyle {}K}
K
{\displaystyle {}K}
inhomogeneous linear system
x
+
y
=
1
y
+
z
=
0
x
+
y
+
z
=
0
.
{\displaystyle {\begin{matrix}x&+y&&=&1\\&y&+z&=&0\\x&+y&+z&=&0\,.\end{matrix}}}
Show by an example that the linear system given by three equations I, II, III is not equivalent to the linear system given by the three equations I-II, I-III, II-III.
Hand-in-exercises
Solve the following system of inhomogeneous linear equations.
x
+
2
y
+
3
z
+
4
w
=
1
2
x
+
3
y
+
4
z
+
5
w
=
7
x
+
z
=
9
x
+
5
y
+
5
z
+
w
=
0
.
{\displaystyle {\begin{matrix}x&+2y&+3z&+4w&=&1\\2x&+3y&+4z&+5w&=&7\\x&\,\,\,\,\,\,\,\,&+z&\,\,\,\,\,\,\,\,&=&9\\x&+5y&+5z&+w&=&0\,.\end{matrix}}}
Consider in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
E
=
{
(
x
,
y
,
z
)
∈
R
3
∣
3
x
+
4
y
+
5
z
=
2
}
and
F
=
{
(
x
,
y
,
z
)
∈
R
3
∣
2
x
−
y
+
3
z
=
−
1
}
.
{\displaystyle E={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 3x+4y+5z=2\right\}}{\text{ and }}F={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 2x-y+3z=-1\right\}}.}
Determine the intersecting line
E
∩
F
{\displaystyle {}E\cap F}
Determine a linear equation for the plane in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
(
1
,
0
,
2
)
,
(
4
,
−
3
,
2
)
,
and
(
2
,
1
,
−
1
)
{\displaystyle (1,0,2),\,\,(4,-3,2),\,{\text{ and }}\,(2,1,-1)}
lie.
Find a polynomial
f
=
a
+
b
X
+
c
X
2
{\displaystyle {}f=a+bX+cX^{2}\,}
with
a
,
b
,
c
∈
C
{\displaystyle {}a,b,c\in \mathbb {C} }
f
(
i
)
=
1
,
f
(
1
)
=
1
+
i
,
f
(
1
−
2
i
)
=
−
i
.
{\displaystyle f({\mathrm {i} })=1,\,f(1)=1+{\mathrm {i} },\,f(1-2{\mathrm {i} })=-{\mathrm {i} }.}
We consider the linear system
2
x
−
a
y
=
−
2
a
x
+
3
z
=
3
−
1
3
x
+
y
+
z
=
2
{\displaystyle {\begin{matrix}2x&-ay&&=&-2\\ax&&+3z&=&3\\-{\frac {1}{3}}x&+y&+z&=&2\end{matrix}}}
over the real numbers, depending on the parameter
a
∈
R
{\displaystyle {}a\in \mathbb {R} }
a
{\displaystyle {}a}
