1. Let A be an m x n matrix. Show that:(a) If x € N (AT A), then Ax e R(A) N N (AT).(b) N (A" A) = N(A).(c) A and A"A have the same rank.(d) If A has linearly independent columns, then AT A is nonsingular.

Let A be an m×n matrix. Show that: (a) If x∈NATA, then Ax∈RA∩NAT (b) NATA=NA. (c) A and AT have…

Answered: Let A be an m x n matrix. Show that: (…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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a1 = ? a2 = ? a3 = ?
3. Let the matrix of L: R³ →R² with respect to the bases S = and T-{]-[-]} T= W₁ W₂ a. Find [L(v₁)T[L(V₂)]T [L(V3)]T b. Find L(v₁), L(V₂), L(V3) c. Find L 12 --CO be A= ( (100)
det(A?) = 8. 1693619 B det (;A) = 2. u10121 1101210007 1693619210 det(A – B) = -2. u10121 %3D 101 0007- 16 210 det(B") = 210 det(B¯'A") = 01210007 A and B 3x3 matrixes and determinant(A)=4 and determinant(B)=6 . for the options given above whats true? 19210 1012 0121 e Message 12:10 (09:28) II 2/3
15. Given matrix N = . Find the value of x such that NN' 1, where I is the 2x2 identity matrix. (A) 3 3 (C) 2 (В) 3 (D)
Prove if y(t): = y₁ (t) Y₂(t) such that y(0) = [ (9) dy satisfy = dt = Ay; t > 0 ; where A is 2 x 2 constant matrix with real entries satisfying trace(A = 0 and Det(A) > 0. Then y₁ (t) and y₂(t) both are oscillating functions of t.
Determine eAt by first finding a fundamental matrix X(t) for x'=Ax and then using the formula At = X(t)X(0)¯ ¹. [21] 25 1 A = First, find X(t). Choose the correct answer below. O A. X(t) = B. X(t) = O c. X(t)= D. X(t)= Next, find et eAt= = e-4t e6t 5e6t 5e 4t -4t - e6t 5e6t - 4t 5e (1 – t)e-4 e6t (5+1)e-4t 5e6t - 4t 6t sin 6t cos 6t 5e6t sin 6t cos 6t 4t 5e
7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: Rª → R¹ by T(x) = Ax?
I need the answer as soon as possible
where A = Ife is defined as e¹=1+A+ *].0<*. , 0
Q2: convolve the following two signals using matrix method: x(n) = 2n[u[n) + u(n = 3)] 1 h(n) = sin(n) [u(n + 2) - u(n-1)]
a) T : R² → R², T(H) = 1]₁ T([]) = [] T ¹([₁]) - b) Find a 2x2 matrix |C 25 = [2²3] Previous
First digit is 2 last digit is 8

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Let A be an m x n matrix. Show that: ( a ) Ifxε Ν (Α'Α), then AxE R(Α) Ω Ν (A"). (b) N (AT A) = N(A). (c) A and AT A have the same rank.