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The projection of a vector onto the image of a set of vectors { _1, _2 } is eq…
Related Topics
Wize University Linear Algebra Textbook > Orthogonality
Orthogonal Complement and Orthogonal Projection (COMING SOON)
3 Activities
The projection of a vector
b
⃗
\vb
b
onto the image of a set of vectors
{
v
⃗
1
,
v
⃗
2
}
\big\{ \vv_1, \vv_2 \big\}
{
v
1
,
v
2
}
is equal to the sum of the projection of the vector
b
⃗
\vb
b
onto each of the individual vectors,
i.e.
:
proj
⇀
Im
(
v
⃗
1
,
v
⃗
2
)
(
b
⃗
)
=
proȷ
⇀
v
1
⃗
(
b
⃗
)
+
proȷ
⇀
v
2
⃗
(
b
⃗
)
\overrightharpoon{\textrm{proj}}_{_{\textrm{Im}(\vv_1,\vv_2)}}(\vb) = \proj{b}{v_1} + \proj{b}{v_2}
proj
Im
(
v
1
,
v
2
)
(
b
)
=
pro
ȷ
v
1
(
b
)
+
pro
ȷ
v
2
(
b
)
Answer
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Check Submission
More Orthogonal Complement and Orthogonal Projection (COMING SOON) Questions:
19.4F_Final_Builder_Ch_17.3_Least_Squares_$\tkco{eg4}$_$\key{Final}$_Builder_$\tkcth{17.3.}\tkcf{4}$_
The figure below illustrates the relationship between the
Im
(
A
‾
)
\bct{\text{Im}\, (\A)}
Im
(
A
)
and
b
⃗
\bco{\color{cyan}\vb}
b
for the system given by:
{
x
−
2
y
=
−
4
2
x
−
4
y
=
7
⇔
[
1
−
2
2
−
4
]
[
a
b
c
]
⏟
:
=
A
‾
[
x
y
]
=
[
−
4
7
]
[
a
b
c
]
⏟
:
=
b
⃗
\left\{ \begin{array}{rr} x - 2y \quad =& -4 \\ 2x - 4y \quad =& 7 \end{array} \right. % \qquad \Leftrightarrow \qquad % \underbrace{ \begin{bmatrix} \; 1 & -2 \; \\ \; 2 & -4 \; \end{bmatrix} \vphantom{\colvecth{a}{b}{c}\!\!\!\!} }_{:= \, \bct{\A}} % \colvec{x}{y} % = % \underbrace{ \colvec{-4}{7} \vphantom{\colvecth{a}{b}{c}} }_{:=\, \bco{\color{cyan}\vb}}
{
x
−
2
y
=
2
x
−
4
y
=
−
4
7
⇔
:=
A
[
1
2
−
2
−
4
]
a
b
c
[
x
y
]
=
:=
b
[
−
4
7
]
a
b
c
Let
b
⃗
Im
(
A
‾
)
=
proj
⇀
Im
(
A
‾
)
(
b
⃗
)
\vb_{_{\textrm{Im}(\A)}} = \overrightharpoon{\text{proj}}_{_{\textrm{Im}(\A)}}(\vb)
b
Im
(
A
)
=
proj
Im
(
A
)
(
b
)
,
i.e.
the (vector) component of
b
⃗
\vb
b
which is in the image of
A
‾
\A
A
. Identify this vector on the figure; we'll call this component
b
⃗
∣
∣
\vb_{_{||}}
b
∣∣
for short.
Span
Let
B
:
=
{
(
1
,
1
,
0
,
2
)
,
(
0
,
1
,
0
,
−
1
)
,
(
0
,
0
,
1
,
1
)
}
B:=\{(1,1,0,2),(0,1,0,-1),(0,0,1,1)\}
B
:=
{(
1
,
1
,
0
,
2
)
,
(
0
,
1
,
0
,
−
1
)
,
(
0
,
0
,
1
,
1
)}
, and
U
:
=
s
p
a
n
B
U:=spanB
U
:=
s
p
an
B
.
a)
Find a pairwise orthogonal set of nonzero vectors
O
O
O
so that
U
=
s
p
a
n
O
U=spanO
U
=
s
p
an
O
.
b)
Write the Fourier expansion of
(
0
,
0
,
1
,
0
)
(0,0,1,0)
(
0
,
0
,
1
,
0
)
in terms of the basis
O
O
O
.
19.4F_Final_Builder_Ch_17.2_Orthogonal_Complement_$\tkco{eg4}$_$\key{Final}$_Builder_$\tkcth{17.2.}\tkcf{4}$_
Suppose
U
=
{
v
⃗
1
,
…
,
v
⃗
n
}
U = \left\{ \vv_1,\, \dots\,,\, \vv_n \right\}
U
=
{
v
1
,
…
,
v
n
}
is an orthogonal basis for
R
n
\mathbb{R}^{n}
R
n
, then any vector
w
⃗
\vw
w
in
R
n
\mathbb{R}^{n}
R
n
can be written as a linear combination of these vectors,
i.e.
w
⃗
=
∑
i
n
c
i
v
⃗
i
=
c
1
v
⃗
1
+
c
2
v
⃗
2
+
…
+
c
n
v
⃗
n
.
\vw = \sum_{i}^{n} c_i \, \vv_i = c_1 \, \vv_1 + c_2 \, \vv_2 + \, \dots \, + c_n \, \vv_n.
w
=
i
∑
n
c
i
v
i
=
c
1
v
1
+
c
2
v
2
+
…
+
c
n
v
n
.
Find an expression for the constants
c
1
c_1
c
1
,
c
2
c_2
c
2
,
…
\dots
…
,
c
n
c_n
c
n
in terms of
w
⃗
\vw
w
and the vectors in
U
U
U
.
19.4F_WML_6_$\tkco{eg10}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{11}$_
Find the set of all vectors perpendicular to the vectors
{
a
⃗
1
,
a
⃗
2
}
=
{
[
1
1
0
1
]
,
[
0
1
1
1
]
}
\left\{ \va_1,\, \va_2\ \right\} = \left\{ \colvecf{1}{1}{0}{1}, \, \colvecf{0}{1}{1}{1} \right\}
{
a
1
,
a
2
}
=
⎩
⎨
⎧
1
1
0
1
,
0
1
1
1
⎭
⎬
⎫
Consider the subspace
W
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
−
y
+
3
z
=
0
}
W=\Big\{(x,y,z)\in \mathbb{R}^3 \; | \; x-y+3z=0 \Big\}
W
=
{
(
x
,
y
,
z
)
∈
R
3
∣
x
−
y
+
3
z
=
0
}
of
R
\mathbb{R}
R
.
Find a basis for
W
W
W
that contains the vector
v
⃗
=
[
0
3
1
]
\vec{v}=[0 \; 3 \; 1]
v
=
[
0
3
1
]
.
Find orthogonal complement of
W
W
W
in
R
3
\mathbb{R}^3
R
3
.
19.4F_WML_6_$\tkco{eg7}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{7}$_
Find a basis for the set of all vectors that are perpendicular to both
{
[
1
1
0
2
]
,
[
1
0
1
0
]
}
\left\{ \colvecf{1}{1}{0}{2}, \, \colvecf{1}{0}{1}{0} \right\}
⎩
⎨
⎧
1
1
0
2
,
1
0
1
0
⎭
⎬
⎫
19.4F_WML_6_$\tkco{eg13}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{14}$_
Show that
any
vector
v
⃗
\vv
v
in the span of the set
V
=
{
v
⃗
1
,
v
⃗
2
,
v
⃗
3
}
=
{
[
1
0
2
]
,
[
2
1
4
]
,
[
1
1
2
]
}
V = \left\{ \vv_1,\, \vv_2,\, \vv_3 \right\} = \left\{ \colvecth{1}{0}{2}, \, \colvecth{2}{1}{4}, \, \colvecth{1}{1}{2} \right\}
V
=
{
v
1
,
v
2
,
v
3
}
=
⎩
⎨
⎧
1
0
2
,
2
1
4
,
1
1
2
⎭
⎬
⎫
will be perpendicular to any vector
w
⃗
\vw
w
in the span of the orthogonal complement
W
=
V
⊥
W = V^{\perp}
W
=
V
⊥
of the set.
19.4F_WML_6_$\tkco{eg12}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{13}$_
Describe the orthogonal complement of the matrix whose columns are given by the vectors
{
v
⃗
1
,
v
⃗
2
,
v
⃗
3
}
=
{
[
1
0
2
]
,
[
2
1
4
]
,
[
1
1
2
]
}
\left\{ \vv_1,\, \vv_2,\, \vv_3 \right\} = \left\{ \colvecth{1}{0}{2}, \, \colvecth{2}{1}{4}, \, \colvecth{1}{1}{2} \right\}
{
v
1
,
v
2
,
v
3
}
=
⎩
⎨
⎧
1
0
2
,
2
1
4
,
1
1
2
⎭
⎬
⎫
by finding a basis for the orthogonal complement, and describing the span of the basis in geometric terms.
Example: Fourier Expansion
Example: Fourier Expansion
Write the vector
(
1
,
1
,
1
)
(1,1,1)
(
1
,
1
,
1
)
as a linear combination of
{
(
1
,
0
,
1
)
,
(
1
,
1
,
−
1
)
,
(
−
1
,
2
,
1
)
}
\{(1,0,1),(1,1,-1), (-1,2,1)\}
{(
1
,
0
,
1
)
,
(
1
,
1
,
−
1
)
,
(
−
1
,
2
,
1
)}
Basis of Orthogonal Complement
Find an orthonormal basis for the orthogonal complement of the intersection of the planes
2
x
+
y
−
z
=
0
2x+y-z=0
2
x
+
y
−
z
=
0
and
4
x
−
3
y
=
0
4x-3y=0
4
x
−
3
y
=
0
in
R
3
\mathbb{R}^3
R
3
.
Orthogonal Complements
Suppose
U
U
U
is a subspace of
R
n
\mathbb{R}^n
R
n
. Prove that
(
U
⊥
)
⊥
=
U
(U^\perp)^\perp=U
(
U
⊥
)
⊥
=
U
.
Orthogonal Complement
If
U
U
U
is a subspace of
R
n
\mathbb{R}^n
R
n
, show that
d
i
m
(
U
)
+
d
i
m
(
U
⊥
)
=
n
dim(U)+dim(U^\perp)=n
d
im
(
U
)
+
d
im
(
U
⊥
)
=
n
.
Basis of Orthogonal Complement
Find a basis for the orthogonal complement of the intersection of the planes
2
x
+
y
−
z
=
0
2x+y-z=0
2
x
+
y
−
z
=
0
and
4
x
−
3
y
=
0
4x-3y=0
4
x
−
3
y
=
0
in
R
3
\mathbb{R}^3
R
3
.
Projection onto a subspace
Suppose
u
⃗
,
v
⃗
∈
R
n
\vec u,\vec v\in\mathbb{R}^n
u
,
v
∈
R
n
satisfy
u
⃗
⋅
v
⃗
=
0
,
∣
∣
u
⃗
∣
∣
=
∣
∣
v
⃗
∣
∣
=
1
\vec u\cdot\vec v=0, ||\vec u||=||\vec v||=1
u
⋅
v
=
0
,
∣∣
u
∣∣
=
∣∣
v
∣∣
=
1
, and define
U
:
=
s
p
a
n
{
u
⃗
,
v
⃗
}
U := span\{\vec u,\vec v\}
U
:=
s
p
an
{
u
,
v
}
.
Using the definition that
p
r
o
j
U
x
⃗
=
A
(
A
T
A
)
−
1
A
T
x
⃗
proj_U\vec x=A(A^TA)^{-1}A^T\vec x
p
r
o
j
U
x
=
A
(
A
T
A
)
−
1
A
T
x
, where
A
=
[
u
⃗
v
⃗
]
A=\left[\begin{array}{rr}\vec u&\vec v\end{array}\right]
A
=
[
u
v
]
, show that for all
x
⃗
∈
R
n
\vec x\in\mathbb{R}^n
x
∈
R
n
we have that
p
r
o
j
U
x
⃗
=
p
r
o
j
u
⃗
x
⃗
+
p
r
o
j
v
⃗
x
⃗
proj_U\vec x=proj_{\vec u}\vec x+proj_{\vec v}\vec x
p
r
o
j
U
x
=
p
r
o
j
u
x
+
p
r
o
j
v
x
.
Projection onto a Subspace
Suppose
S
:
=
{
a
⃗
1
,
.
.
.
,
a
⃗
k
}
S:=\{\vec a_1,...,\vec a_k\}
S
:=
{
a
1
,
...
,
a
k
}
is a set of orthonormal vectors in
R
n
\mathbb{R}^n
R
n
,
U
:
=
s
p
a
n
(
S
)
U:=span(S)
U
:=
s
p
an
(
S
)
, and suppose that the
n
×
k
n\times k
n
×
k
matrix
A
A
A
is defined by
A
:
=
[
a
⃗
1
a
⃗
2
.
.
.
a
⃗
k
]
A:=\left[\begin{array}{rrrr}\vec a_1&\vec a_2&...&\vec a_k\end{array}\right]
A
:=
[
a
1
a
2
...
a
k
]
.
Show that
A
A
T
AA^T
A
A
T
is the matrix of the linear transformation:
R
n
→
R
n
x
⃗
→
p
r
o
j
U
x
⃗
\begin{array}{l}\mathbb{R}^n\rightarrow\mathbb{R}^n\\ \vec x\rightarrow proj_U\vec x\end{array}
R
n
→
R
n
x
→
p
r
o
j
U
x
.