Let
With addition of functions and scalar multiplication defined as in Example
Example
Verify that
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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
- Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.arrow_forwardProofProve in full detail that M2,2, with the standard operations, is a vector space.arrow_forwardIn Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 39.arrow_forward
- Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}arrow_forwardProve that in a given vector space V, the additive inverse of a vector is unique.arrow_forwardTake 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.arrow_forward
- In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V= F, W=finF:f(0)=1arrow_forwardTrue or False? In Exercises 49 and 50, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a To show that a set is not a vector space, it is sufficient to show that just one axiom is not satisfied. b The set of all first-degree polynomials with the standard operations is a vector space. c The set of all pairs of real numbers of the form (0,y), with the standard operations on R2, is a vector space.arrow_forwardProve that in a given vector space V, the zero vector is unique.arrow_forward
- Testing for a Vector Space In Exercises 13-36, determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all first-degree polynomial functions ax,a0, whose graphs pass through the origin.arrow_forwardIdentify the zero element and standard basis for each of the isomorphic vector spaces in Example 12. EXAMPLE 12 Isomorphic Vector spaces The vector spaces below are isomorphic to each other. a. R4=4space b. M4,1=spaceofall41matrices c. M2,2=spaceofall22matrices d. P3=spaceofallpolynomialsofdegree3orless e. V={(x1,x2,x3,x4,0):xiisarealnumber} subspace of R5arrow_forwardLet u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.arrow_forward
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