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19.4F_Final_Builder_Ch_17.2_Orthogonal_Complement_$\tkco{eg1}$_$\key{Final}$_Bu…

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Wize University Linear Algebra Textbook > Orthogonality

Orthonormal Basis and Gram-Schmidt Process

5 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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).

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 .
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}