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Wize University Linear Algebra Textbook > Orthogonality

Orthonormal Basis and Gram-Schmidt Process

5 Activities

Wize University Linear Algebra Textbook > Orthogonality

Orthogonal Complement and Orthogonal Projection (COMING SOON)

3 Activities

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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)\} , and U:=spanBU:=spanB .

a) Find a pairwise orthogonal set of nonzero vectors OO so that U=spanOU=spanO .

b) Write the Fourier expansion of (0,0,1,0)(0,0,1,0) in terms of the basis OO .

c) Can(0,0,1,0)(0,0,1,0) be written as a linear combination of elements of BB ?

d) What is the distance between (0,0,1,0)(0,0,1,0) and the closest vector in UU to (0,0,1,0)(0,0,1,0) ?
More Orthonormal Basis and Gram-Schmidt Process Questions:
19.4F_Final_Builder_Ch_17.2_Orthogonal_Complement_$\tkco{eg1}$_$\key{Final}$_Builder_$\tkcth{17.2.}\tkcf{1}$_$\key{Quiz}$
Show that the set of vectors
{[121][−11−1][−303]} \left\{ \colvecth{1}{2}{1} \colvecth{-1}{1}{-1} \colvecth{-3}{0}{3} \right\}
is a basis for R3\mathbb{R}^3using *only* the fact that the vectors are all perpendicular to one another (i.e. no points would be awarded for using Gaussian elimination).
Consider the subspace W={(x,y,z)∈R3  ∣  x−y+3z=0}W=\Big\{(x,y,z)\in \mathbb{R}^3 \; | \; x-y+3z=0 \Big\} of R\mathbb{R}.
Find a basis for WW that contains the vector v⃗=[0  3  1]\vec{v}=[0 \; 3 \; 1].
Find orthogonal complement of WW in R3\mathbb{R}^3.
Gram-Schmidt process
Consider the basis B={(1,3),(2,1)}B=\{(1,3),(2,1)\} of R2\mathbb{R}^2
Use the Gram-Schmidt process to change BB into an orthonormal basis B′B'
Gram-Schmidt process
Use the Gram-Schmidt proces to find an orthonormal basis for U=Span{(1,2,1,2),(3,3,3,4),(7,6,5,3)}U=\text{Span}\{(1,2,1,2),(3,3,3,4),(7,6,5,3)\}
Practice: Gram-Schmidt Process
Use the Gram-Schmidt process to create an orthonormal basis for the subspace WW of R4\mathbb{R}^4:
W=Span({u⃗1,u⃗2,u⃗3})W=\text{Span}\Big(\{\vec u_1,\vec u_2,\vec u_3\}\Big) where u⃗1,u⃗2,u⃗3\vec u_1,\vec u_2, \vec u_3 are the columns of the matrix:
A=[136112134110]A=\left[\begin{array}{rrr} 1&3&6\\ 1&1&2\\ 1&3&4\\ 1&1&0 \end{array}\right]
Practice Question: Gram-Schmidt
Find an orthonormal basis of null(A)null(A) , where A=[−12312−45−1]A=\left[\begin{array}{rrrr}-1&2&3&1\\2&-4&5&-1\end{array}\right] .
Orthonormal Basis and Gram-Schmidt Process
Example: Gram-Schmidt Process
Let U=span{(2,1,−1),(−5,1,3),(1,1,9)}U=\text{span}\{(2,1,-1),(-5,1,3),(1,1,9)\} be a subspace of R3\mathbb{R}^3.
Use the Gram-Schmidt process to find an orthonormal basis for UU.
Orthonormal Basis
Example: Orthonormal Sets
Part A)
Show that {(1,0,1,0),(0,1,0,1),(−1,0,1,0),(0,−1,0,1)}\{(1,0,1,0),(0,1,0,1),(-1,0,1,0),(0,-1,0,1)\} is an orthogonal set.
Basis of Orthogonal Complement
Find an orthonormal basis for the orthogonal complement of the intersection of the planes 2x+y−z=02x+y-z=0 and 4x−3y=04x-3y=0 in R3\mathbb{R}^3 .
If we apply the Gram-Schmidt process to the vectors (3,0,4,0)(3, 0, 4, 0), (−1,0,7,0)(−1, 0, 7, 0), and (0,1,3,0)(0, 1, 3, 0) in that order, to obtain the orthonormal vectors u1,u2,u3u_1, u_2, u_3 in that order, then what isu3u_3?
Projection onto a Subspace
Calculate projspan{(1,1,0,1),(1,0,1,1)}(4,−2,−1,−1)proj_{span\{(1,1,0,1),(1,0,1,1)\}}(4,-2,-1,-1) .
B={[11−1],[101],[1−2−1]}B=\left\{\begin{bmatrix}1\\1\\-1\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\-2\\-1\end{bmatrix}\right\} is an orthogonal basis of R3R^3 . Which of the following is the coordinate vector of [311]\begin{bmatrix}3\\1\\1\end{bmatrix}relative to the basis?
Find an orthogonal basis for R3R^3containing the vector [12−2]\begin{bmatrix}1\\2\\-2\end{bmatrix}
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)} and b⃗\bco{\color{cyan}\vb} for the system given by:
{x−2y=−42x−4y=7⇔[  1−2    2−4  ][abc] ⁣ ⁣ ⁣ ⁣⏟:= A‾[xy]=[−47][abc]⏟:= b⃗ \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}}
Let b⃗Im(A‾)=proj⇀Im(A‾)(b⃗)\vb_{_{\textrm{Im}(\A)}} = \overrightharpoon{\text{proj}}_{_{\textrm{Im}(\A)}}(\vb), i.e. the (vector) component of b⃗\vb which is in the image of A‾\A. Identify this vector on the figure; we'll call this component b⃗∣∣\vb_{_{||}} for short.
The projection of a vector b⃗\vb onto the image of a set of vectors {v⃗1,v⃗2}\big\{ \vv_1, \vv_2 \big\} is equal to the sum of the projection of the vector b⃗\vb onto each of the individual vectors, i.e.:
proj⇀Im(v⃗1,v⃗2)(b⃗)=proȷ⇀v1⃗ ⁣(b⃗)+proȷ⇀v2⃗ ⁣(b⃗) \overrightharpoon{\textrm{proj}}_{_{\textrm{Im}(\vv_1,\vv_2)}}(\vb) = \proj{b}{v_1} + \proj{b}{v_2}
19.4F_Final_Builder_Ch_17.2_Orthogonal_Complement_$\tkco{eg4}$_$\key{Final}$_Builder_$\tkcth{17.2.}\tkcf{4}$_
Suppose U={v⃗1, … , v⃗n}U = \left\{ \vv_1,\, \dots\,,\, \vv_n \right\} is an orthogonal basis for Rn\mathbb{R}^{n}, then any vector w⃗\vw in Rn\mathbb{R}^{n} can be written as a linear combination of these vectors, i.e.
w⃗=∑inci v⃗i=c1 v⃗1+c2 v⃗2+ … +cn v⃗n. \vw = \sum_{i}^{n} c_i \, \vv_i = c_1 \, \vv_1 + c_2 \, \vv_2 + \, \dots \, + c_n \, \vv_n.
Find an expression for the constants c1c_1, c2c_2, …\dots, cnc_n in terms of w⃗\vw and the vectors in UU.
19.4F_WML_6_$\tkco{eg10}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{11}$_
Find the set of all vectors perpendicular to the vectors {a⃗1, a⃗2 }={[1101], [0111]}\left\{ \va_1,\, \va_2\ \right\} = \left\{ \colvecf{1}{1}{0}{1}, \, \colvecf{0}{1}{1}{1} \right\}
Consider the subspace W={(x,y,z)∈R3  ∣  x−y+3z=0}W=\Big\{(x,y,z)\in \mathbb{R}^3 \; | \; x-y+3z=0 \Big\} of R\mathbb{R}.
Find a basis for WW that contains the vector v⃗=[0  3  1]\vec{v}=[0 \; 3 \; 1].
Find orthogonal complement of WW in R3\mathbb{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 {[1102], [1010]}\left\{ \colvecf{1}{1}{0}{2}, \, \colvecf{1}{0}{1}{0} \right\}
19.4F_WML_6_$\tkco{eg13}$_$\key{Final}$_Builder_$\tkcth{17.1.}\tkct{14}$_
Show that any vector v⃗\vv in the span of the set
V={v⃗1, v⃗2, v⃗3}={[102], [214], [112]} V = \left\{ \vv_1,\, \vv_2,\, \vv_3 \right\} = \left\{ \colvecth{1}{0}{2}, \, \colvecth{2}{1}{4}, \, \colvecth{1}{1}{2} \right\}
will be perpendicular to any vector w⃗\vw in the span of the orthogonal complement W=V⊥W = V^{\perp} 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 {v⃗1, v⃗2, v⃗3}={[102], [214], [112]}\left\{ \vv_1,\, \vv_2,\, \vv_3 \right\} = \left\{ \colvecth{1}{0}{2}, \, \colvecth{2}{1}{4}, \, \colvecth{1}{1}{2} \right\} 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) as a linear combination of {(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 2x+y−z=02x+y-z=0 and 4x−3y=04x-3y=0 in R3\mathbb{R}^3 .
Orthogonal Complements
Suppose UU is a subspace of Rn\mathbb{R}^n . Prove that (U⊥)⊥=U(U^\perp)^\perp=U .
Orthogonal Complement
If UU is a subspace of Rn\mathbb{R}^n , show that dim(U)+dim(U⊥)=ndim(U)+dim(U^\perp)=n .
Basis of Orthogonal Complement
Find a basis for the orthogonal complement of the intersection of the planes 2x+y−z=02x+y-z=0 and 4x−3y=04x-3y=0 in R3\mathbb{R}^3 .
Projection onto a subspace
Suppose u⃗,v⃗∈Rn\vec u,\vec v\in\mathbb{R}^n satisfy u⃗⋅v⃗=0,∣∣u⃗∣∣=∣∣v⃗∣∣=1\vec u\cdot\vec v=0, ||\vec u||=||\vec v||=1, and define U:=span{u⃗,v⃗}U := span\{\vec u,\vec v\} .
Using the definition that projUx⃗=A(ATA)−1ATx⃗proj_U\vec x=A(A^TA)^{-1}A^T\vec x , where A=[u⃗v⃗]A=\left[\begin{array}{rr}\vec u&\vec v\end{array}\right] , show that for all x⃗∈Rn\vec x\in\mathbb{R}^n we have that projUx⃗=proju⃗x⃗+projv⃗x⃗proj_U\vec x=proj_{\vec u}\vec x+proj_{\vec v}\vec x .
Projection onto a Subspace
Suppose S:={a⃗1,...,a⃗k}S:=\{\vec a_1,...,\vec a_k\} is a set of orthonormal vectors in Rn\mathbb{R}^n , U:=span(S)U:=span(S) , and suppose that the n×kn\times k matrix AA is defined by A:=[a⃗1a⃗2...a⃗k]A:=\left[\begin{array}{rrrr}\vec a_1&\vec a_2&...&\vec a_k\end{array}\right] .
Show that AATAA^T is the matrix of the linear transformation: Rn→Rnx⃗→projUx⃗\begin{array}{l}\mathbb{R}^n\rightarrow\mathbb{R}^n\\ \vec x\rightarrow proj_U\vec x\end{array} .