Let V be a finite dimensional vector space with dimension n > 0. Let S = {V1, V2, Vn} be a family of vectors in V. Prove that the following three conditions are ... equivalent (1) S is a basis for V. (2) S is linearly independent. (3) S spans V.

Let V be a finite dimensional vector space with dimension n > 0. Let S = {V1, V2, Vn} be a family of vectors in V. Prove that the following three conditions are ... equivalent (1) S is a basis for V. (2) S is linearly independent. (3) S spans V.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: 0. -6 2. be an orthogonal basis for the vector space R°with the Euclidean inner product (i.e. dot…

Q: Let B=[u, be a base for V=R², where = (1,2) and =(-1,1). We know that every vector in V is a linear…

Q: Consider following three sets: (i) set A has p vectors of the vector space V, and A= span{V}, (ii)…

Q: Let 1 A = 0 2 Find a pair of vectors u, v in R4 whose span is the kernel of A. U= 1 0 1 0 -2 -1 0 1…

A: The Solution is done.Explanation:Approach to solving the question: Detailed explanation: Examples:…

Q: Consider the following vectors in R³: 5 12 7 -7 11 Find a basis of Span (V₁, V2, V3, V4, V5). V1 =…

A: Given That : Vectors in R5 are v1=5127-7-11 , v2=102414-14-22 , v3=76-11-13 , v4=-3-30-1-5 ,…

Q: Let B = {b1,.…, b,} be a basis for a vector space V, and k > n. Prove that any set of vectors…

Q: Determine whether or not the given vectors in R² form a basis for R². 5 V₁ [] 9 Do the given vectors…

Q: 14. Let P₂ be the vector space of polynomials of degree ≤ 2. Let D = {1 + t, 1 − 5t, 2 + t²}. Check…

Q: 16. 0 1 BOL 9 0 1 6 - 1 2 9 5 −3 3 -4 - - 9 3

A: Given Data: Let us consider the given data, 1001, -21-11, 6-12-1 ,5-33-4,03-11 To Find: The basis…

Q: Let V be an n-dimensional vector space, and let W = {v1, v2,..., Vn} be a subset of V. Consider the…

A: Let V be a n dimensional vector space and Let W=v1,v2,...,vn be a subset of V. Its given that the…

Q: . Let V be a vector space, and suppose that {V₁,...,Vn} is V. Explain why any basis for V must have…

Q: Consider the basis D = {(1, 1, -1), (2, 1, 3), (-1, 2, 1)} of R3 and the vector v = (1, 1, 1) in R3.…

Q: Let the set of non-zero vectors V1, V2,..., Vn in some vector space V be orthonormal: (Vi, V) = dij.…

Q: Consider the three basic vectors: (2,1,0,-1),(-1,1,1,1),(2,7,4,1) a. give two vectors that are in…

A: Given the three basic vectors: (2,1,0,-1),(-1,1,1,1),and (2,7,4,1) We have…

Q: C. Determine if the following set of vectors is orthogonal: [-3 1 2] T, [2 4 1]T, [1 -1 2] T. Then,…

Q: V is a vector space with bases B = (vị, V2, V3) and B' = (w1, w2, w3). The change of basis matrix…

A: V is vector space with basis B={v1,v2,v3} and B'=w1,w2,w3 The change of basis matrix from basis B to…

Q: (c) Let V and W be given as above. By using linear extension method, determine whether V is…

Q: Prove the following result: Let V be a finite-dimensional vector space, let V₁, V2, ..., Un be…

Q: Determine whether or not the given vectors in R² form a basis for R². 6 - [8]/1/2 5 8 Do the given…

A: Let c1 v1 +c2 v2 =0.So 6c1 +5c2 =0 and 3c1 +8c2 =0.Solving these we get c1 =c2 =0.

Q: 2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the…

Q: Let the basis vectors be given by a = [3,1] and 6 = [3,3]. Give the image of a general coordinate…

Q: What is the dimension of P₂, the vector space of all polynomials of degree at most two? b. Find the…

Q: (a) i. Define what it means for vectors v₁,..., n of a vector space V to be linearly dependent. ii.…

A: Vectors in a vector space are linearly dependent if there exist constants (not all zero) such…

Q: Prove that if the set of vectors {ν1, . . . , νn} is linearly independent in V, then so is the list…

Q: 7. Let V = R2 and let S be the set of vectors S = {V1,V2} -3 Is S is linearly independent? Explain…

Q: Find the new coordinate vector for the vector x after performing the specified change of basis.…

Q: be a vector expressed in coordinates with respect to the standard basis of R. Let 9. Let 10 be a…

A: In the given question, concept of basis is applied. Basis If every element of V can be expressed in…

Q: Let W = span{(1,1,-1), (2,3,1), (5,6, -2)}. i Find a basis for W. ii. Determine whether the vector X…

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

6.
Let V be a finite dimensional vector space with dimension n > 0. Let S =
{V1, V2, ·..
Vn} be a family of vectors in V. Prove that the following three conditions are
equivalent
(1) S is a basis for V.
(2) S is linearly independent.
(3) S spans V.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

(b) Is the following set of vectors S basis for vector space V? Where (i) S = {(1,2), (0, 3), (1, 5)} and V = R? (ii) S = {(-1,3, 2), (6, 1, 1)} and V = R³ If not explain the reason.
4. Let S = {v,,v2, V3, V4, V5} where, 3 1 2 3 3 ,V2 9. ,V3 2 la -1 4 Find a basis for the space span(S) which is a subset of S. Then express each vector that is not in the basis as a linear combination of the basis vectors.
1. Determine whether the vectors (1,-1) and (1, 1)T are linearly inde- pendent or not. 2. Show that the vectors (1,0,0), (1, 1, 0) and (1,1,1) form a basis of R³. 3. Suppose that the vectors (1, 2, -1,0) and (1,3,2,0) span the sub- space U of R4. Determine whether the vector (1, 1, 1, 0) belongs to U or not.
Suppose now we have a set of non standard basis vectors v1, v2, .., Un e R" b) which satisfies the following equation: 1 1 WV = I = [v1, v2, ..., vn] where w and vi are the rows and columns of W and V, respectively. Any vector z € R" can be written as a linear combination of the st andard basis vectors v1, v2, ..., Vn as follows: k=1 Determine the coefficients ak in terms of z and wf only.
3. Let T = 1 7 W= 1 and x = 0 (a) Are vectors u, V, and w linearly independent? (b) Do u, v', and w span the entire R³? (c) Do u, v', and w form a basis of R³? abc po
E11 ,E12 , E21 and En Use the 7. to show B= {E; E12, E21, E22} is a basis definitions provided on the first page for M2(R).
(b) Let W = M2x1(C) be a vector space over C. Give a basis B for W.