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Elementary Linear Algebra (MindTap Course List)

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 61CR: Find the bases for the four fundamental subspaces of the matrix. A=[010030101].

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Give integers p and q such that Nul A is a subspace of RP and Col A is a subspace of R9.
-5 7
A =
-3 0
4 1
Nul A is a subspace of RP for p =
and Col A is a subspace of Rª for q =

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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?
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.
Consider the vector spaces P0,P1,P2,...,Pn where Pk is the set of all polynomials of degree less than or equal to k, with standard operations. Show that if jk, then Pj is the subspace of Pk.
Find the bases for the four fundamental subspaces of the matrix. A=[010030101].
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.
(a) find the orthogonal complement S⊥, and (b) find the direct sum S⊕S⊥. S is the subspace of R5 consisting of all vectors whosethird and fourth components are zero.
(a) find k such that Nul A is a subspace of ℝk, and (b) find k such that ColA is a subspace of ℝk,A = [1 -3 9 0 -5]
(a) find the orthogonal complement S⊥, and (b) find the direct sum S⊕S⊥. S is the subspace of R3 consisting of the xz-plane.
Let S=span(e1), T=span(e2) and W=span(e1+e3) be subspaces of R3. S is orthogonal to T, T is orthogonal to W, then S is orthogonal to W.
Which of the given subsets of R4 are subspaces? The set of all vectors of the form a) [a b c d], where a = 0 and b = −d b) [a b c d], where a = 1, b = 0 and a + d = 1

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