Let q be a unit vector in !Rn and let W be the subspace spanned by q. Show that the orthogonal projection of a vector v onto W is given by projw(v) = (qqT)v and that the matrix of this projection is thus qq T.

Let q be a unit vector in !Rn and let W be the subspace spanned by q. Show that the orthogonal projection of a vector v onto W is given by projw(v) = (qqT)v and that the matrix of this projection is thus qq T.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 26EQ

See similar textbooks

Related questions

Q: Let W be the subspace of IRª spanned by the vectors 4 2 and -1 Find the matrix A of the orthogonal…

A: Consider the given information: Let W be the subspace of ℝ4 spanned by vectors :1-11-1,4-2-1-7 To…

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Using the orthogonal projection of vector Y to get the component of y along the vector space W.

Q: Find a basis for the subspace of R^4 that is spanned by the vectors v1 = (1, 1, 1, 1), v2 = (2,2, 2,…

A: Let us consider set S then the set S is linearly independent or dependent Determine the linear…

Q: Find the dimension of the subspace H of IR2 spanned by the vectors and

Q: Find the orthogonal projection of u on the subspace of R³ spanned by the vectors v₁ and v₂. u = (1,…

Q: let S be the collection of vector [x] (3x1 matrix) in R3 that satisfy the given…

Q: Let W be the subspace of R4 spanned by the vectors 1 1 and -6 -1 6 1 4 Find the matrix A of the…

Q: Find the standard matrix for the orthogonal projection of R3 onto the subspace spanned by aj=(1,1)…

A: The vectors that span the subspace are a1=1, 1 and a2=0, 1. The normalising vectors are 12, 12 and…

Q: Find the projection of the vector v onto the subspace S. S = span

Q: If u and v are vectors in R and W is a subspace of R with (u - v) belong to W, then both u and v…

A: If u and v are vectors in R3 and W is a subspace of R3 with (u-v) belong to W, then both u and v…

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: We need to find y.

Q: Find the orthogonal projection ŷ of y = onto the subspace W = Span { uị u2 5 Ex: 1.23 : ŷ =

Q: . Let u₁ = (1, -2, -1), U₂ = (2, 2,-2), and v = = (0, 0, 1). v is not in the subspace W spanned by…

A: In this question, a vector that is orthogonal to a given vector is to be determined. A non-zero…

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: In this question, the concept of the Orthogonal Decomposition theorem is applied. The vector…

Q: 2 Find the orthogonal projection of u = 1 onto the subspace of R3 spanned by the vectors -3 - 2 V1 =…

Q: The set S = ( (x.y, 2) ER |2 = 2, x,y,ZER)is a subspace under vector addition and scalar…

A: This question is related to linear algebra topic vector space.

Q: For a certain subspace W, the orthogonal projection of the vector y = 3 onto W is ŷ 2. Use this…

Q: 2. Show that U = {(x,y, x) : , y E R} is a subspace of R$. Then, find a linear complement of U in R.…

Q: Find the orthogonal projection of 3 onto the subspace W of R* spanned by 1 1 -1 1 -1 1 1 1 [1] 1 II…

Q: Find a basis for the subspace of R, the plane 2x +y-z+ 3w-0 and convert it into an orthonormal ban

A: Given the subspace 2x+y-z+3w=0. To get the orthonormal basis. The formula for orthonormal basis is,…

Q: Let u, = 2 |, u2 and uz = 0 Note that u, and uz are orthogonal but that uz is not orthogonal to u,…

A: The orthogonal vector is given as follows :

Q: Let V be a vector space and U1,u2,U3,Ug are vectors in V. What is the dimension of the subspace S=…

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors ui

Q: Find the projection of the vector v onto the subspace S.

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

Q: Let W be the subspace of R spanned by the vectors -2 and Find the matrix A of the orthogonal…

A: Given that W be the subspace of ℝ2 spanned by the vectors 1-21 and…

Q: Let V be the subspace of C[a, b] spanned by1, ex, e−x, and let D be the differentiation operatoron…

Q: Is W a subspace of the vector space? W is the set of all matrices in Mn,n with zero determinants.

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

Q: Suppose W is a subspace of Rn spanned by n nonzero orthogonal vectors. Explain why W = Rn.

Q: Find the projection matrix P that projects vectors in R onto W Let W be the subspace of R spanned by…

A: The inverse of a 2×2 matrix is calculated using the formula: a b c d -1=1deta b c d d -b -c a…

Q: Find the orthogonal projection of onto the subspace W of Rª spanned by projw() = v= 4 -2 -13] 3 15…

Q: Find the orthogonal projection of V = onto the subspace W of R' spanned by U1 = , U2 Uz = VW ||

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

Q: Let u, 1₂" orthogonal to u, and u The nonzero vector v and u Note that u, and u, are orthogonal. It…

Q: Find the dimension of the subspace W of R4 spanned by S = {v1, v2, v3} = {(−1, 2, 5, 0), (3, 0, 1,…

A: Dimension of a subspace W is given by the number of linearly independent vectors in the basis of W,…

Q: Let U and W be subspaces of R³ for which dimU=1, dimW=2, and UgW. Show that R=UOW.

A: I have shown all the conditions of direct sum of two subspaces

Q: Let u, = u2 = -1. and uz =0| Note that u, and uz are orthogonal but that uz is not orthogonal to…

A: Introduction: If two vectors are perpendicular to one other, they are said to be orthogonal. The dot…

Q: Find the distance from the point x = (1, 5, –4) of R³ to the subspace W consisting of all vectors of…

A: First we will find out the orthogonal basis for the subspace W. Then we calculate the orthogonal…

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Given vectors are y=466−1, u1=1101, u2=−120−1 and u3=−1011. We have to write the vector y as a sum…

Q: Find the orthogonal projection ŷ of the vector y = onto the subspace W = Span u 2 Ex: 5 C ŷ = %3D

Q: If B is basis for (x, t) then By = {BnY/BE basis for the subspace topology Ty on y IS 9

Q: For a certain subspace W, the orthogonal projection of the vector y = 5 onto W 4 is ŷ = |1. Use this…

Q: Find the orthogonal projection ŷ of y = 3 onto the subspace W = Span { u1 = , U2 = 3 Ex: 1.23 %3D

Q: Find the orthogonal projection of onto the subspace W of Rª spanned by 1 projw (v)

Q: Find a basis for the subspace of R° consisting of all vectors perpendicular to v, that is find a…

Q: Find the orthogonal compliment of the subspace W of 2x2 upper triangular matrices in the vector…

A: Using standard inner product in the space of matrices.

Q: Consider the vector spaceV = M3x3(C) over the field R. Let U be the subspace of V consisting of all…

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors u;. are orthogonal.)…

Question

Let q be a unit vector in !Rn and let W be the subspace spanned by q. Show that the orthogonal projection of a vector v onto W is given by proj_w(v) = (qq^T)v and that the matrix of this projection is thus qq ^T.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Linear Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.
Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?
In Exercises 1-4, let S be the collection of vectors in [xy]in2 that satisfy the given property. In each case either prove that S forms a subspace of 2 or give a counterexample to show that it does not. xy0
In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=Mnn, W is the set of diagonal nn matrices