Write in
Q
2
{\displaystyle {}\mathbb {Q} ^{2}}
(
2
,
−
7
)
{\displaystyle (2,-7)}
as a linear combination of the vectors
(
5
,
−
3
)
and
(
−
11
,
4
)
.
{\displaystyle (5,-3){\text{ and }}(-11,4).}
Write in
C
2
{\displaystyle {}\mathbb {C} ^{2}}
(
1
,
0
)
{\displaystyle (1,0)}
as a
linear combination
of the vectors
(
3
+
5
i
,
−
3
+
2
i
)
and
(
1
−
6
i
,
4
−
i
)
.
{\displaystyle (3+5{\mathrm {i} },-3+2{\mathrm {i} }){\text{ and }}(1-6{\mathrm {i} },4-{\mathrm {i} }).}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .
Let
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
w
∈
V
{\displaystyle {}w\in V}
w
,
v
i
,
i
∈
I
,
{\displaystyle w,v_{i},i\in I,}
is a system of generators of
V
{\displaystyle {}V}
w
{\displaystyle {}w}
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
Let
K
{\displaystyle {}K}
field ,
and let
V
{\displaystyle {}V}
K
{\displaystyle {}K}
vector space .
Prove the following facts.
Let
U
j
{\displaystyle {}U_{j}}
j
∈
J
{\displaystyle {}j\in J}
linear subspaces
of
V
{\displaystyle {}V}
U
=
⋂
j
∈
J
U
j
{\displaystyle {}U=\bigcap _{j\in J}U_{j}\,}
is a subspace.
Let
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
W
{\displaystyle {}W}
V
{\displaystyle {}V}
W
{\displaystyle {}W}
V
{\displaystyle {}V}
The family
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
⟨
v
i
,
i
∈
I
⟩
=
V
.
{\displaystyle {}\langle v_{i},\,i\in I\rangle =V\,.}
Show that the three vectors
(
0
1
2
1
)
,
(
4
3
0
2
)
,
(
1
7
0
−
1
)
{\displaystyle {\begin{pmatrix}0\\1\\2\\1\end{pmatrix}},\,{\begin{pmatrix}4\\3\\0\\2\end{pmatrix}},\,{\begin{pmatrix}1\\7\\0\\-1\end{pmatrix}}}
in
R
4
{\displaystyle {}\mathbb {R} ^{4}}
linearly independent .
Give an example of three vectors in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
Let
K
{\displaystyle {}K}
V
{\displaystyle {}V}
K
{\displaystyle {}K}
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
If the family is linearly independent, then for each subset
J
⊆
I
{\displaystyle {}J\subseteq I}
v
i
{\displaystyle {}v_{i}}
i
∈
J
{\displaystyle {}i\in J}
The empty family is linearly independent.
If the family contains the null vector, then it is not linearly independent.
If a vector appears several times in the family, then the family is not linearly independent.
A vector
v
{\displaystyle {}v}
v
≠
0
{\displaystyle {}v\neq 0}
Two vectors
v
{\displaystyle {}v}
u
{\displaystyle {}u}
u
{\displaystyle {}u}
v
{\displaystyle {}v}
Let
K
{\displaystyle {}K}
V
{\displaystyle {}V}
K
{\displaystyle {}K}
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
λ
i
{\displaystyle {}\lambda _{i}}
i
∈
I
{\displaystyle {}i\in I}
≠
0
{\displaystyle {}\neq 0}
K
{\displaystyle {}K}
v
i
{\displaystyle {}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
V
{\displaystyle {}V}
V
{\displaystyle {}V}
λ
i
v
i
{\displaystyle {}\lambda _{i}v_{i}}
i
∈
I
{\displaystyle {}i\in I}
Determine a basis for the solution space of the linear equation
3
x
+
4
y
−
2
z
+
5
w
=
0
.
{\displaystyle {}3x+4y-2z+5w=0\,.}
Determine a basis for the solution space of the linear system of equations
−
2
x
+
3
y
−
z
+
4
w
=
0
and
3
z
−
2
w
=
0.
{\displaystyle -2x+3y-z+4w=0{\text{ and }}3z-2w=0.}
Prove that in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
(
2
1
5
)
,
(
1
3
7
)
,
(
4
1
2
)
{\displaystyle {\begin{pmatrix}2\\1\\5\end{pmatrix}}\,,{\begin{pmatrix}1\\3\\7\end{pmatrix}}\,,{\begin{pmatrix}4\\1\\2\end{pmatrix}}}
form a
basis.
Establish if in
C
2
{\displaystyle {}\mathbb {C} ^{2}}
(
2
+
7
i
3
−
i
)
and
(
15
+
26
i
13
−
7
i
)
{\displaystyle {\begin{pmatrix}2+7{\mathrm {i} }\\3-{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}15+26{\mathrm {i} }\\13-7{\mathrm {i} }\end{pmatrix}}}
form a basis.
Let
K
{\displaystyle {}K}
field .
Find a linear system of equations in three variables whose solution space is exactly
{
λ
(
3
2
−
5
)
∣
λ
∈
K
}
.
{\displaystyle {\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.}
Hand-in-exercises
Write in
Q
3
{\displaystyle {}\mathbb {Q} ^{3}}
(
2
,
5
,
−
3
)
{\displaystyle (2,5,-3)}
as a linear combination of the vectors
(
1
,
2
,
3
)
,
(
0
,
1
,
1
)
,
and
(
−
1
,
2
,
4
)
.
{\displaystyle (1,2,3),(0,1,1),{\text{ and }}(-1,2,4).}
Prove that it cannot be expressed as a linear combination of two of the three vectors.
Establish if in
R
3
{\displaystyle {}\mathbb {R} ^{3}}
(
2
3
−
5
)
,
(
9
2
6
)
,
(
−
1
4
−
1
)
{\displaystyle {\begin{pmatrix}2\\3\\-5\end{pmatrix}}\,,{\begin{pmatrix}9\\2\\6\end{pmatrix}}\,,{\begin{pmatrix}-1\\4\\-1\end{pmatrix}}}
form a basis.
Establish if in
C
2
{\displaystyle {}\mathbb {C} ^{2}}
(
2
−
7
i
−
3
+
2
i
)
and
(
5
+
6
i
3
−
17
i
)
{\displaystyle {\begin{pmatrix}2-7{\mathrm {i} }\\-3+2{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}5+6{\mathrm {i} }\\3-17{\mathrm {i} }\end{pmatrix}}}
form a basis.
