Show that the given set V is closed under addition and multiplication by scalars and is therefore a subspace of R³.V is the set of all y such that 9x = 4y.---Zx+ua+b= y+v (Simplify your answer.)z+wNext, describe the relationship between the first and second components of each of the two vectors a and b.9x = 4y9u 4vAdd the left sides of the two equations given in the previous step, and add the right sides.9(x+u) = 4(y + v)(Type your answer in factored form.)Based on the result of the previous step, is the vector sum a + b in V?equal totimes its second component.The vector sum a + b ▼an element of the set V because times its first component(Type an integer or a fraction.)XLet c be a scalar. Recall that a = y. Find the scalar multiple ca.Zca =(Simplify your answer.)Next, determine the relationship between the first and second components of ca.9(cx) = c(x) = c(y) = (cy)Based on the result of the previous step, is the scalar multiple ca in V?The scalar multiple caan element of the set V becauseequal totimes its second component.(Type an integer or a fraction.)How does the analysis above show that V is a subspace of R³?Since▼is/are elements of V, the set V▼closed under multiplication by scalars and is therefore a subspace of R³.times its first componentclosed under addition and

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Answered: Show that the given set V is closed…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 42CR: 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...

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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 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 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 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?
Find the projection of the vector v=[102]T onto the subspace S=span{[011],[011]}.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Find the bases for the four fundamental subspaces of the matrix. A=[010030101].
Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.
Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.
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

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Show that the given set V is closed under addition and multiplication by scalars and is therefore a subspace of R³. V is the set of all y such that 9x = 4y. x+u