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Span
Related Topics
Wize University Linear Algebra Textbook > Vector Spaces and Subspaces
Span
4 Activities
Practice: Span
Let
u
⃗
=
⟨
1
,
0
,
1
⟩
,
v
⃗
=
⟨
1
,
1
,
1
⟩
,
w
⃗
=
⟨
1
,
−
1
,
1
⟩
\vec u = \lang 1,0,1\rang,\ \vec v = \lang 1,1,1\rang,\ \vec w = \lang 1,-1,1 \rang
u
=
⟨
1
,
0
,
1
⟩
,
v
=
⟨
1
,
1
,
1
⟩
,
w
=
⟨
1
,
−
1
,
1
⟩
be vectors in
R
3
\reals^3
R
3
, and suppose
b
⃗
=
⟨
3
,
y
,
z
⟩
∈
span
(
{
u
⃗
,
v
⃗
,
w
⃗
}
)
\vec b = \lang 3,y,z \rang \in \text{span}(\{\vec u, \vec v, \vec w\})
b
=
⟨
3
,
y
,
z
⟩
∈
span
({
u
,
v
,
w
})
.
Find the value of
z
z
z
.
z
=
z=
z
=
I don't know
Check Submission
More Span Questions:
Practice: Span
Show that for all
w
⃗
,
x
⃗
,
y
⃗
,
z
⃗
∈
R
n
\vec w,\vec x,\vec y, \vec z\in\mathbb{R}^n
w
,
x
,
y
,
z
∈
R
n
, if
w
⃗
∈
S
p
a
n
{
x
⃗
,
y
⃗
,
z
⃗
}
\vec w\in Span\{\vec x, \vec y, \vec z\}
w
∈
S
p
an
{
x
,
y
,
z
}
then
w
⃗
∈
S
p
a
n
{
x
⃗
−
y
⃗
,
y
⃗
−
z
⃗
,
z
⃗
}
\vec w\in Span\{\vec x-\vec y, \vec y-\vec z,\vec z\}
w
∈
S
p
an
{
x
−
y
,
y
−
z
,
z
}
.
When is a Vector in the Span of Others
Determine if
x
⃗
=
(
16
,
−
15
,
8
)
∈
Span
{
u
⃗
,
v
⃗
,
w
⃗
}
\vec x=(16,-15, 8) \in \text{Span}\{\vec u, \vec v, \vec w\}
x
=
(
16
,
−
15
,
8
)
∈
Span
{
u
,
v
,
w
}
if:
u
⃗
=
(
1
,
1
,
1
)
,
v
⃗
=
(
1
,
1
,
0
)
,
w
⃗
=
(
1
,
0
,
−
1
)
\vec u = (1,1,1), \vec v = (1,1,0), \vec w =(1,0,-1)
u
=
(
1
,
1
,
1
)
,
v
=
(
1
,
1
,
0
)
,
w
=
(
1
,
0
,
−
1
)
Practice: Span
Let
u
⃗
=
(
2
,
2
,
1
)
,
v
⃗
=
(
2
,
1
,
2
)
,
w
⃗
=
(
2
,
5
,
−
2
)
,
x
⃗
=
(
x
,
y
,
z
)
\vec u = (2,2,1), \vec v = (2,1,2), \vec w = (2,5,-2), \vec x = (x,y,z)
u
=
(
2
,
2
,
1
)
,
v
=
(
2
,
1
,
2
)
,
w
=
(
2
,
5
,
−
2
)
,
x
=
(
x
,
y
,
z
)
be vectors in
R
3
\mathbb{R}^3
R
3
Find the conditions that the values
x
,
y
,
z
x,y,z
x
,
y
,
z
must satisfy in order for
x
⃗
\vec x
x
to be in
Span
(
{
u
⃗
,
v
⃗
,
w
⃗
}
)
\text{Span}(\{\vec u, \vec v, \vec w\})
Span
({
u
,
v
,
w
})
Span
Practice: Span
Let
u
⃗
=
⟨
1
,
0
,
1
⟩
,
v
⃗
=
⟨
1
,
1
,
1
⟩
,
w
⃗
=
⟨
1
,
−
1
,
1
⟩
\vec u = \lang 1,0,1\rang,\ \vec v = \lang 1,1,1\rang,\ \vec w = \lang 1,-1,1 \rang
u
=
⟨
1
,
0
,
1
⟩
,
v
=
⟨
1
,
1
,
1
⟩
,
w
=
⟨
1
,
−
1
,
1
⟩
be vectors in
R
3
\reals^3
R
3
, and suppose
b
⃗
=
⟨
3
,
y
,
z
⟩
∈
span
(
{
u
⃗
,
v
⃗
,
w
⃗
}
)
\vec b = \lang 3,y,z \rang \in \text{span}(\{\vec u, \vec v, \vec w\})
b
=
⟨
3
,
y
,
z
⟩
∈
span
({
u
,
v
,
w
})
.
Find the value of
z
z
z
.
Span
Select all of the vectors that are in
Span
(
u
⃗
,
v
⃗
)
\text{Span}(\vec u, \vec v)
Span
(
u
,
v
)
where
u
⃗
=
(
−
1
,
1
,
2
)
,
v
⃗
=
(
2
,
1
,
−
3
)
\vec u=(-1,1,2), \vec v=(2,1,-3)
u
=
(
−
1
,
1
,
2
)
,
v
=
(
2
,
1
,
−
3
)
Linear Independence and Span concepts
Which of the following statements are true about the set of vectors
{
(
1
,
1
,
−
1
,
0
)
,
(
−
1
,
3
,
−
1
,
1
)
,
(
2
,
0
,
−
3
,
4
)
}
\{(1,1,-1,0),(-1,3,-1,1),(2,0,-3,4)\}
{(
1
,
1
,
−
1
,
0
)
,
(
−
1
,
3
,
−
1
,
1
)
,
(
2
,
0
,
−
3
,
4
)}
?
133 - FML 3 - 18.1W - e.g. 31
Knowing that
[
0
6
3
3
0
−
7
1
5
1
−
1
1
3
]
→
RREF
[
1
0
0
0
1
0
0
0
1
0
0
0
]
\bcb{\begin{bmatrix}%[c] 0 & 6 & 3 \\ 3 & 0 & -7 \\ 1 & 5 & 1 \\ -1 & 1 & 3 \end{bmatrix} % \xrightarrow{\text{{\tiny RREF}}} % \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}}
0
3
1
−
1
6
0
5
1
3
−
7
1
3
RREF
1
0
0
0
0
1
0
0
0
0
1
0
,
determine whether the vector
v
⃗
1
=
<
0
,
3
,
1
,
−
1
>
\bcb{\vec{v}_1 = \big< 0,3,1,-1\big>}
v
1
=
⟨
0
,
3
,
1
,
−
1
⟩
is in the span of the vectors
v
⃗
2
=
<
6
,
0
,
5
,
1
>
\bcb{\vec{v}_2 = \big< 6,0,5,1\big>}
v
2
=
⟨
6
,
0
,
5
,
1
⟩
and
v
⃗
3
=
<
3
,
−
7
,
1
,
3
>
\bcb{\vec{v}_3 = \big< 3,-7,1,3\big>}
v
3
=
⟨
3
,
−
7
,
1
,
3
⟩
.
Consider the space of polynomials of degree 3 or lower,
P
3
\mathcal{P}_3
P
3
. Give an example of a linearly independent set of vectors whose span is the entirety of
P
3
\mathcal{P}_3
P
3
.
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{6}$_$\key{|Strc>5.2}$
Describe the set of all linear combinations (sums of constant multiples of) the vectors:
v
⃗
=
[
1
1
1
]
\vv = \colvecth{1}{1}{1}
v
=
1
1
1
and
w
⃗
=
[
−
1
2
1
]
\vw = \colvecth{-1}{2}{1}
w
=
−
1
2
1
.
133 - FML 3 - 18.1W - e.g. 30
Given the standard basis vectors for
R
2
\bcb{\mathbb{R}^{2}}
R
2
,
viz.
ı
^
\bcb{\ihat}
^
and
ȷ
^
\bcb{\jhat}
^
and the vector
u
⃗
=
[
0
2
]
\bcb{\vec{u} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}}
u
=
[
0
2
]
, determine whether
ı
^
\bcb{\ihat}
^
is in the
span
{
u
⃗
,
ȷ
^
}
\bcb{ \vspan{\vec{u}, \jhat} }
span
{
u
,
^
}
.
Briefly justify your answer.
Let
(
V
,
+
,
⋅
)
(V, +, \cdot)
(
V
,
+
,
⋅
)
be a real vector space. If
v
1
,
…
,
v
n
v_1, \dots, v_n
v
1
,
…
,
v
n
is a set of vectors that spans the entirety of
V
V
V
, prove that the following set of vectors also spans
V
V
V
.
v
1
−
v
2
,
v
2
−
v
3
,
…
,
v
n
−
1
−
v
n
,
v
n
v_1 - v_2, v_2 - v_3, \dots, v_{n - 1} - v_n, v_n
v
1
−
v
2
,
v
2
−
v
3
,
…
,
v
n
−
1
−
v
n
,
v
n
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{4}$_$\key{|Strc>5.2}$
Describe the set of all constant multiples of the vector:
v
⃗
=
[
−
1
2
1
]
\vv = \colvecth{-1}{2}{1}
v
=
−
1
2
1
.
19.4F_Final_Builder_Ch_4.6_Span_Line_From_FAQ_$\tkco{eg}$_$\key{Final}$_Builder_$\tkcth{4.6.}\tkcf{2}$_$\key{|Strc>5.2}$
Describe the set of all constant multiples of the vector
v
⃗
=
[
1
1
]
\vv = \colvec{1}{1}
v
=
[
1
1
]
.
Practice Question: Arbitrary Vector as Linear Combination
Show that every vector in
R
2
\mathbb{R}^2
R
2
is a linear combination of
(
1
,
−
1
)
(1,-1)
(
1
,
−
1
)
and
(
2
,
1
)
(2,1)
(
2
,
1
)
.
Arbitrary Vector as Linear Combination
Show every vector in
R
3
\mathbb{R}^3
R
3
is a combination of
(
8
,
0
,
0
)
,
(
0
,
1
,
1
)
,
(
0
,
2
,
3
)
(8,0,0),(0,1,1),(0,2,3)
(
8
,
0
,
0
)
,
(
0
,
1
,
1
)
,
(
0
,
2
,
3
)
.
Suppose that
S
=
s
p
a
n
(
[
1
−
1
2
]
,
[
−
3
3
−
6
]
,
[
1
2
1
]
,
[
5
4
7
]
)
S=span\left(\begin{bmatrix}1\\-1\\2\end{bmatrix},\begin{bmatrix}-3\\3\\-6\end{bmatrix},\begin{bmatrix}1\\2\\1\end{bmatrix},\begin{bmatrix}5\\4\\7\end{bmatrix}\right)
S
=
s
p
an
1
−
1
2
,
−
3
3
−
6
,
1
2
1
,
5
4
7
, which of the following statements is true?