5.Let v1, v2, ..., Vn E R". Let A be the n x n matrix whose j-th column is v;(j = 1, 2,n), that is,[v |А —V1V2V.Prove thatV1, V2, ·..,Vnform a basis for R"det (A) + 0.

Answered: Image /qna-images/answer/cb6a3958-d81a-4817-9adc-32b944afa514.jpg

Answered: 5. Let v1, V2, ..., Vn € R". Let A be…

5. Let v1, V2, ..., Vn € R". Let A be then x n matrix whose j-th column is v; (j = 1, 2, n), that is, A = v2 .. V1 Vn Prove that V1, V2, ..., Vn form a basis for R" det (A) # 0.

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: Find the coordinate matrix of x in R" relative to the basis B'. B' = {(6, 0), (0, 2)}, x = (48, 10)

Q: The third column of the change of basis matrix from the basis B = {u= (1,1,1); v= (1,0,1); w=…

Q: Suppose A is the matrix for T: R3 → R3 relative to the standard basis. Find the diagonal matrix A'…

Q: Find the coordinate matrix of x in R^ relative to the basis B'. B' = {(5, 0), (0, 2)}, x = (10, 8)…

A: If A is a basis, then x∈Rn can be written as linear combination of the elements of A. When a scalar…

Q: Q. No. 2 a) Find matrix A such that given set is Col A. ns + (n + 1)t r +s – nt (п+ 2)r + s r, s, t…

Q: Let Find an orthonormal basis of the row space of A. { [ 1 1 A = [424] -2 -3 ]]. [

A: Row space of a matrix is the vector subspace spanned by the rows.

Q: 3. Give an example of an orthonormal set with 4 non-trivial vectors in R5.

A: We have to solve given problem:

Q: Let B = {A1, A2, A3, A4} and B2 = {C1, C2. Cs, C4} when : %3D %3D A3 A4 = 0 0 C; - (; }).c- (? :).c-…

A: By Bartleby policy I supposed to solve first 3 subparts.But I am solving all subparts.

Q: 4. (a) Let L be the line in R' spanned by the vector (1,1, 1, 1). Find a basis for the orthogonal…

Q: Denote the column vectors of the matrix 11,922.. , ann +] True, false, or uncertain: a; is…

Q: 1. In the {li), j), Ik)} representation, the normalized basis vectors are chosen as li) = (1). 1) =…

Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

Q: tind the equation of a polynomial Shose graph, passes through the points t-1,7), (0,2)(2;4)

A: As per guidelines I had solved the first question. Thank you.

Q: The second column of the matrix of change basis from the basis B = {u= (1,1,1); v = (1,0,1); w=…

Q: 1 3 Let the inner product in R be generated by the matrix 4 = -1 2 Calculate the inner product of…

Q: Generate the matrix A of order n= 100, whose generic element is a; = sin(max(i, j)) The condition…

Q: 8. Which of the following are orthonormal s 1 A = {U₁ = (₂0₁2), U₂=(2,0, B = { U₁ = (2,0, ₂2), U₂ =…

Q: Find the coordinate matrix of x in Rº relative to the basis B'. B' = {(3, 0), (0, 8)}, x = (24, 16)…

A: Given the basis B'=3,0,0,8 and x=24,16 Let x1, x2 is coordinate vector that is xB'=x1x2 Then it…

Q: Find the coordinate matrix of x relative to the orthonormal basis B in R". 3 10 -), (o, 1, 0), 0, 10…

A: To solve the following problem and find the coordinate matrix of x

Q: The second column of the matrix of change of basis from the basis B={u=(1,1,1); v= (1,0,1);…

A: What is Change of Basis: Any element of a vector space of finite dimension n can be uniquely…

Q: Proposition 11.5.1. If A and B are is defined, then matrices such that the matrix product (AB)=BT…

Q: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

A: We are given that the non-zero vectors s(1), s(2), ..., s(n) are conjugate to a positive definite…

Q: 1s) La L : A → R be the limsir operator Lip(z)) – p(0)r*+27/(c). Find the matrix A representing with…

Q: 22. Find a basis for R4 that contains the vectors (3, 2, 3, 3) and (5, 4, 5, 5)

Q: (2 0 2 Consider the matrix A = 0 2 1. Find a Jordan matrix J and an 1 -2 0, invertible matrix Q such…

A: The given matrix: A= 2020211-20 The Jordan form of the matrix is given by finding the eigenvalues…

Q: 10. Let (1,2,3) be a solution for the vector equation ru + yv + zw = 0 and consider the matrix…

Q: 7) Find the Jacobian matrix at The print (1,2,1) of The Willowing Funchun: soudainish and Nidevorsh…

Q: 2 COLAS -1 Consider the two bases B= for R² and C= P √2 a.) Compute the change-of-coordinates matrix…

Q: Let A be a 9x6 matrix with nullify equal to 1 a) find the rank and nullify of A^T b) suppose that…

Q: Q2) If the complex nx n matrix A(t) is continuous on an interval I, then the solution of the system…

A: What is Vector Space: A vector space is a collection of elements referred to as vectors. There are…

Q: Let wi = w2 = and set B = (w1,w2). Then B is a basis of R². (C 9). If T : R? → R² has matrix with…

A: We have to find first general formula for T the the sum of the coefficient

Q: Let A be an m x n matrix with orthonormal columns, and x, y be in R" then prove tat the linear…

Q: The second column of the matrix of change of basis from the basis B={u= (1,1,1); v = (1,0,1); w=…

A: Since the basis is changed from B to B', the second column: s can be written as a linear combination…

Q: 9. Consider the orthonormal basis S for R³ S = (v₁,v₂,v,} where v₁ = (0,1,0) and v₁ = Let u = V₂ = =…

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

5.
Let v1, v2, ..., Vn E R". Let A be the n x n matrix whose j-th column is v;
(j = 1, 2,
n), that is,
[v |
А —
V1
V2
V.
Prove that
V1, V2, ·..,
Vn
form a basis for R"
det (A) + 0.

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 3

VIEW

Step by step

Solved in 3 steps with 4 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

Which of the following is true or false? a)Let A be nxn matrix. The equality <Ax,y>=<x,A^Ty> is satisfied with the scalar product in R^n b)f(x)=1 and g(x)=2x−1 are orhogonal vectors in C[0,1] with the standard inner product. c)If x and y are unit vectors in R^n and ∣∣x^Ty∣∣=1 then from cosine theorem we have x and y are linearly independent.
Let B={ }].[],:)} 3 2 1,U} be basis for IR? 2 Find [V]a where 15 6.
Let CER" be a matrix such that xCx > 0 for all nonzero vectors & Prove that for k= 1,...,n, the leading principal submatrix C, satisfies y all nonzero vectors y ERK. Consequently, each C is nonsingular. R" Cky > 0 for
Find the coordinate matrix of x relative to the orthonormal basis B in R". x = (10, 20, 5), B = {G) (-), (0, o, 1)} [x]g = 1 1
Find the matrix A' for T relative to the basis B'. T: R² A' = R², T(x, y) = (x + y, 3y), B' = {(-4, 1), (1, -1)}
1. Which of the following vectors is not a linear combination of and 1» [ ] 3 2): 3) [2] '4) 5) all of them 2. Suppose A is an m x n matrix, x ɛ R" and Ax = b. Then b is in 1) R" | 2) R*T 3) R™†ªT4) R*n 3. Consider the statements below. (I) For each b in R" the equation Ax = b has a solution. (II) Each b in R" is a linear combination of the columns of A. (III) The columns of A span Rm. (IV) A has a pivot in each row. Which one of the four statements is not equivalent to the others? ( 1) I] 2) II| 3) I | 4) IV
6. Let |V(12)) be a state on H") 8 H2). Let {|k)} be an orthonormal basis for H") and {\j)} an orthonormal basis for H2). Show that |V(12)) = Ik) ® (k|V(12)) and
3. orthonormal basis B generated by B.
-13/27] -4/27 -4/9 5/9 -20/27] -2/27 Let v₁ = √2 = and √3 = Find a vector in IR4 such that -20/27 ' ' -2/9 17/27 4/9 -2/3 2/9 {1, 2, 3, 4} is an orthonormal set. Answer:
suppose that x↓1,..,x↓(n-1) are linearly independent vectors in R^n and also x↓n is notin the span of x↓1,..,x↓(n-1) . 1. prove that x↓1,...,x↓n is a basis of R^n. 2. Prove that if x↓(n+1)∈R^n, then x↓1,..,x↓n,x↓n+1 are not linearly independent
33. Prove that the row vectors of an n x n invertible matrix A form a basis for Rn.
A3) Lat M = (sin(nz): n1,3,5,} Can M form a basis for L'(0, n)?