Explanation Approach: Basis of a vector space W is defined as a subset { w 1 , w 2 , ... , w n } of vectors in W that are linearly independent and span W . Normal equation is given by ( v − w * ) w i = 0 , where i = 1 , 2 , ... , n . Two vectors a and b are said to be orthogonal vectors if their dot product is 0 i.e. a ⋅ b = 0 Calculation: Given, W = R ( B ) , B = [ 1 2 1 1 0 1 ] W = { w : w = x 1 [ 1 1 0 ] + x 2 [ 2 2 1 ] ; x 1 , x 2 ∈ ℝ } = S p a n { [ 1 1 0 ] , [ 2 2 1 ] } = S p a n { w 1 , w 2 } For w 1 = [ 1 1 0 ] and w 2 = [ 2 2 1 ] . Checking for linear dependence of w 1 and w 2 Let, α w 1 + β w 2 = 0 α [ 1 1 0 ] + β [ 2 2 1 ] = 0 [ α + 2 β α + 2 β β ] = 0 Which is only true for α = 0 and β = 0 Thus, w 1 and w 2 are independent, Thus Basis of W ( B W ) = { w 1 , w 2 } Find approximation w * : Let u 1 = w 1 , We choose a scalar a so that u 2 = w 2 + a u 1 is orthogonal to u 1

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

See similar textbooks

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

Videos

Question

Chapter 3.9, Problem 14E

To determine

To Find:

The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w, and verify that v−w is orthogonal to the basis vectors for v=[3,3,3]T.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.9, Problem 13E

Chapter 3.9, Problem 15E

Students have asked these similar questions

Enter the matrix PD+B: b) Let v be a vector in R³ whose vector of coordinates relative to the basis B is given by 0 [V] B Find the vector [v] of coordinates of v relative to the basis of D. Enter the vector [v] in the form [C₁, C2, C3]: =

Q2) Find the coordinate vector [x]g of x relative to the basis B={b1, b2}: b2 =

Find the new coordinate vector for the vector x after performing the specified change of basis. 6) Consider two bases B and C such that b1 = c₁ - 5c2 and b2 = 2c₁ + 6c2. Find [x]c. Suppose x = b1 + 3b2. That is, suppose [x]B =

Chapter 3 Solutions

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 4E Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 6E Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Exercises 1-11 refer to the vectors given in 1....Ch. 3.1 - Prob. 10E

Knowledge Booster

$Algebra$

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.

The basis for the subspace W . For the given vector v solve the normal equations and determine the best approximation w * , and verify that v − w * is orthogonal to the basis vectors for v = [ 3 , 3 , 3 ] T .

Solution Summary: The author explains the basis of a vector space W.

Concept explainers

Videos

To Find:

The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w, and verify that v−w is orthogonal to the basis vectors for v=[3,3,3]T.

Want to see the full answer?

Chapter 3 Solutions

The basis for the subspace W . For the given vector v solve the normal equations and determine the best approximation w * , and verify that v − w * is orthogonal to the basis vectors for v = [ 3 , 3 , 3 ] T .

Solution Summary: The author explains the basis of a vector space W.

Concept explainers

Videos

To Find: The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w*, and verify that v−w* is orthogonal to the basis vectors for v=[3,3,3]T.

Want to see the full answer?

Chapter 3 Solutions

To Find:

The basis for the subspace W. For the given vector v solve the normal equations and determine the best approximation w, and verify that v−w is orthogonal to the basis vectors for v=[3,3,3]T.