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Orthonormal Basis

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Wize University Linear Algebra Textbook > Orthogonality

Orthonormal Basis and Gram-Schmidt Process

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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.

Part B)

Normalize each vector to obtain an orthonormal set.
More Orthonormal Basis and Gram-Schmidt Process Questions:
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)\} , 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 .
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.
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}