(3) Problem 3. Please provide a typewritten solution! I will be very grateful!. Please Solve only (a) and b(iv) Problem 3. (a) Let A be an m x n matric. Define what is meant by thenull space of A, N(A). Prove that N{A) is a subspace of R".(b) Let c1, c2, Cz denote the columns of the matrix1-75-816A --15-12-3 -12(i) Find rank(A).(ії) Find a basis for N(A).(iii) Find a basis for the range of A, R(A).(iv) Let d = c1+c2+C3. Before attempting to solve the system, erplainwhy you know that Axsolution of the system of equations Ax = d.d is consistent. Write down a general

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Answered: Problem 3. Please provide a typewritten…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 3AEXP

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1. Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector subspace of all 2 x 2 matrices?
Solve: (a) If a 7 x 9 matrix A has rank 5, what are the dimensions of the fourfundamental subspaces of A? (b) If a 3 x 4 matrix A has rank 3, what are the dimensions of R(A) and N(AT)?
2. For each matrix A of Problem 1, write down orthonormal base for all four fundamental subspaces.(This can be read off from your answers to Problem 1.)
Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}
Find a basis for the subspace of R3 spanned by S = {(2,3, -1), (1,3, -9), (0,1,5)}
Give an example of a non-zero subspace of R^4 which has dimension > 1 and does not contain any of standard basis vectors e1, e2, e3, e4.
9. Show that P2 (polynomials of degree ≤ 2) is a subspace of P3 (polynomials of degree ≤ 3).
1. For each matrix A, find the singular value decomposition in the matrix form A = UΣVT.a)8 41 13b)1 32 6c)−3 1110 −21 5−4 6d)9 7 10 8−13 1 5 −6e)3 7 1 53 1 7 56 2 2 −22. For each matrix A of Problem 1, write down orthonormal base for all four fundamental subspaces.(This can be read off from your answers to Problem 1.)
Suppose A is 10 by 10 and A2 = 0 (zero matrix). So A multiplies each column of A to give the zero vector. This means that the column space of A is contained in the __ . If A has rank r, those subspaces have dimension r � 10 � r. So the rank is '(' � 5.
Suppose masses m1, m2, m3, m4 are located at positions x1, x2, x3, x4 in a line and connected by springs with constants k12, k23, k34 whose natural lengths of extension are l12, l23, l34. Let f1, f2, f3, f4 denote the rightward forces on the masses, e.g., f1 = k12(x2 - x1 - l12). (a) Write the 4 x 4 matrix equation relating the column vectors f and x. Let K denote the matrix in this equation. (b) What are the dimensions of the entries of K in the physics sense (e.g., mass times tim, distance divided by mass, etc.)? (c) What are the dimensions of det(K), again in the physics sense? (d) Suppose K is given numerical values based on the unit meters, kilograms, and seconds. Now the system is rewritten with a matrix K' based on centimeters, grams, and seconds. What is the relationship of K' to K? What is the relationship of det(K') to det(K)?
Consider all 4 by 4 matrices A that are diagonalized by the same fixed eigenvector matrix X. Show that the A's form a subspace (cA and A1 + A2 have this same X). What is this subspace when X = I? What is its dimension?
10. Find a basis for the Subspace. What is the dimension of W?

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