Question: Let S = {v1, . . . , vn} be a set of vectors in a vector space V with the property that every vector

Let S = {v1, . . . , vn} be a set of vectors in a vector space V with the property that every vector in V can be written as a linear combination of v1, . . . , vn in exactly one way. Prove that S is a basis for V.

Step by Step Solution

★★★★★

3.55 Rating (165 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

We must show that S spans V and that S is linearly independent First since every vector can ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

859-L-A-L-S (2827).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let B be a set of vectors in a vector space V with the property that every vector in V can be written uniquely as a linear combination of the vectors in B. Prove that B is a basis for V.

Let S = {v1, v2, v3} be a set of vectors in a vector space V. Prove that 5 is linearly dependent if and only if one of the vectors in 5 is a linear combination of all the other vectors in 5.

Let S = {v1, v2, ( ( ( ( vk} be a set of vectors in a vectors in a vector space V, and let W be a subspace of V containing S. Show that W contains span s.

Let {v1,..., vn} be a spanning set for the vector space V, and let v be any other vector in V. Show that v, v1, ..., vn are linearly dependent.

1. Decide if the following statement is True or False: Let o = V1, V2, V3 be a basis for a vector space V over C, and let Te S(V) be defined by IT]a = 3 0 Let W be the set of vectors x E V that...

Please help with this question 1. Decide if the following statement is True or False: Let o = v1, V2, v3 be a basis for a vector space V over C, and let TE C(V) be defined by 1 O [T]& = 0 Let W be...

1. Decide if the following statement is True or False: Let o = V1, V2, V3 be a basis for a vector space V over C, and let Te S(V) be defined by IT]a = 3 0 Let W be the set of vectors x E V that...

Please give me your own writting solutions, not other's, thank you! ? 1. Decide if the following statement is True or False: Let a = V1, V2, V3 be a basis for a vector space V over C, and let TE C(V)...

Linear Algebra Final answer only - No Explanation Section 5.5 Orthonormal Sets: Problem 1 (1 point) Find the missing coordinates such that the three vectors form an orthonormal basis for IR" : -0.6...

Which of the following statements are TRUE? Every spanning set S of a vector space V is called a basis for V. DA subset S of vectors of a vector space V is called a basis for V, if span (S) = V and S...

a. Precisely what is microbial death? b. Why does a population of microbes not die instantaneously when exposed to an antimicrobial agent?

1. a.* Mg(s) + MgCl(aq) + AgNO3(aq) + _CaCl(aq) = _AgCl(s) + _CH(g) + _O(g) _CO(g) + _HO(g) _NaO(s) + _HO) _KCIO,(s) Al(s) + HCl(aq) g. CHO(g) + O(g) h. b.* C.* d.* e.* f. i. j. k. Classify the...

i want an example of a question for Migrating from Physical to Virtual Servers

Defi ne entrepreneurship. (33.1)

Given that g(x) = x + 6 / x - 3, find g(-6), g(0), and g(3).

Find the domain of the function. (a) g(x) = x + 9 (b) f(x) = -5 / x + 5 (c) h(x) = 1 / x2 + 2x - 3

Graph the function. (a) f(x) = -2x (b) g(x) = x2 - 1

A stock has a beta of 1.00 and an expected return of 10 percent. A risk-free asset currently earns 2.2 percent. a. What is the expected return on a portfolio that is equally invested in the two...

Steven Company has fixed costs of 3764,278 . The unit selling price, variable cost per unit, and contritution margin per unit for the comparyys two products are 35 follows: whole number. x units of x...

Note: In addition to your solution to each computational problem, you must show the supporting work leading to your solution to receive credit for your answer.