Concept explainers
GOAL Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace.
Which of the subsets of
1.
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Linear Algebra with Applications (2-Download)
Additional Math Textbook Solutions
Intermediate Algebra (8th Edition)
Elementary Algebra
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
College Algebra (Collegiate Math)
College Algebra (5th Edition)
- Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are Pythagorean triples.arrow_forwardCalculus Let W1,W2,W3,W4, and W5 be defined as in Example 5. Show that Wi is a subspace of Wj for ij. Example 5 Subspaces of Functions Calculus Let W5 be the vector space of all functions defined on [0,1], and let W1,W2,W3, and W4 be defined as shown below. W1=setofallpolynomialfunctionsthataredefinedon[0,1]W2=setofallfunctionsthataredifferentiableon[0,1]W3=setofallfunctionsthatarecontinuouson[0,1]W4=setofallfunctionsthatareintegrableon[0,1]ShowthatW1W2W3W4W5andthatWiisasubspaceofWjforij.arrow_forwardSubsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R2 whose components are integers.arrow_forward
- Subsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose components are nonnegative.arrow_forwardSubsets That are Not Subspaces In Exercises 7-20, W is not a subspace of V. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all vectors in R3 whose third component is 1.arrow_forwardDetermine subspaces of Mn,n In Exercises 2936, determine whether the subsetMn,n is a subspace ofMn,nwith the standard operations. Justify your answer. The set of all nn diagonal matricesarrow_forward
- Proof Prove that if S1 and S2 are orthogonal subspaces of Rn, then their intersection consists of only the zero vector.arrow_forwardSubsets That Are Not Subspaces In Exercises 7-20 W is not a subspace of vector space. Verify this by giving a specific example that violates the test for a vector subspace Theorem 4.5. W is the set of all matrices in Mn,n with determinants equal to 1.arrow_forwardLet 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?arrow_forward
- Determining Subspace of Mn,n In Exercises 29-36, determine whether the subset of Mn,nis a subspace of Mn,nwith the standard operations. Justify your answer. The set of all nn matrices A that commute with a given matrix B; that is, AB=BA.arrow_forwardVerifying Subspaces In Exercises 1-6, verify that W is a subspace of V. In each case assume that V has the standard operations. W is the set of all 22 matrices of the form [0ab0] V=M2,2arrow_forwardDetermining Subspace of Mn,n In Exercises 29-36, determine whether the subset of Mn,nis a subspace of Mn,nwith the standard operations. Justify your answer. The set of all nn matrices whose trace is nonzero Recall that the trace of a matrix is the sum of the main diagonal entries of the matrix.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning