Question: Let A be an m x n matrix. Show that (a) If x N(ATA), then Ax is in both R(A) and N(AT). (b) N(ATA)

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

Step by Step Solution

★★★★★

3.32 Rating (176 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Ax RA for all vectors x in R n If x NA T A then A T Ax 0 and h... 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)

949-M-L-A-E (690).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let A be an m x n matrix, B an n x r matrix, and C = AB. Show that (a) N(B) is a subspace of N(C). (b) N(C)- is a subspace of N(B) and, consequently, R(CT) is a subspace of R(BT).

Let A be an m x n matrix of rank r and let |x1,..., xr} be a basis for R(AT). Show that {Ax1,. . . .. . ., Axr] is a basis for R(A).

Let A be an m x n matrix of rank n and let P = A(ATA)-1 AT. (a) Show that Pb - b for every b R(A). Explain this in terms of projections. (b) If b R(A), show that Pb = 0. (c) Give a geometric...

4. For each of the following linear transformations, express it in (5.1.1) O. a. T(x1, x2) = (x1 x2, 2x1 +x2, 5x1 3x2).

please help with these 6 practice problems 1 1 .LetA=[2 a space C(A), the null space N (A), the row space, and the left null space of A. . Find a such that rank A =1; and for this value of oz, nd...

Let X be the design matrix used to find the least-squares line to fit data (x 1 y 1 ),.....(x n, y n ). Use a theorem in Section 6.5 to show that the normal equations have a unique solution if and...

This is in MATLAB 177% + 4. Write a function B = BasisOfNullSpace(A) whose input A is a matrix and whose output B is a matrix whose columns form a basis for the nullspace of A. Use the following...

7. For each of the following statements, determine whether it is True or False and explain your reasoning. (a) (3 points) Let A be an m x n matrix and suppose that the entries in each row of A add up...

(a) Let A be an m x n matrix, where n,m> 2, and y R" such that Ay = 0. Then the columns of A are linearly dependent. (b) The following matrix has four linearly independent columns: -1 3 250-3-2 419-3...

1. [2 points each] Let A be an m x n matrix. Define Ker(A) := { x ( R" | Ax = 0} and Im(A) := {y E R" | there is a x ( R" such that Ax = y }. (a) Show that Ker(A) is a subspace of R" and Im(A) is a...

The units of an item available for sale during the year were as follows: There are 36 units of the item in the physical inventory at December 31. The periodic inventory system is used. Determine the...

Find the Thevenin equivalent of the network below at the terminals A-B in figure. 1 1 21x 1 K

Companies can sell new security issues directly to institutional investors. What would this transaction be known as ? Fixed price offering Tender offer Private placement Exiernal distribution

Project Scope Management. What we know about the work that must be done, and the work that is not included in this project.?

Use the Cauchy inequality to prove that" (a) ( r1 +r2 + . . . + rn )n n (r21 + r22 + . . . + r2n for all ri in R and all n 1. (b) r1r2 + r1r3 + r2r3 r21 + r22 + r23 for all r1, r2,and r3 in R.

(a) Show that X and Y are orthogonal in Rn if and only if || X + Y || = ||X - Y||. (b) Show that X + Y and X - Y are orthogonal in Rn if and only if || X || = || Y ||.

If A is n n, show that every eigenvalue of ATA is nonnegative. [Compute ||AX||2 where X is an eigenvector.]

The doubling time of a population of flies is 4 hours. By what factor does the population increase in 35 hours? By what factor does the population increase in 3weeks? By what factor does the...

Ziege Systems is considering the following independent projects for the coming year. Project Required Investment Rate of Return Risk A $4 million 14% High B 5 million 11.5 High C 3 million 9.5 Low D...

The primary risk of mortgage-backed securities is that the prices of have high volatility that the prices of housing will increase that the government will not able to meet the guarantees on the cash...