Question: Let A be an m x n matrix. Show that every vector v in Rn can be written uniquely as w + u where w

Let A be an m x n matrix. Show that every vector v in Rn can be written uniquely as w + u where w is in the null space of A and u is in the column space of AT.

Step by Step Solution

★★★★★

3.49 Rating (169 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Let v be a vector in R n By Theorem 512a the column spa... 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)

951-M-L-A-L-S (7036).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

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...

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).

(12 marks) (a) (6 marks) Given the matrices A and the reduced echelon form of A below, find: i. a basis for Row(A). What is the dimension of Row(A)? ii. a basis for Col(A). What is the dimension of...

4. (20 points) a) Find a subset of the vectors that forms a basis for the space spanned by the vectors V1 = (1, -1, 5, 2), v2 = (-2, 3, 1, 0), v3 = (9, -10, 34, 14) , v4 = (0, 4, 2, -3) , v5 = (-3,...

12:39 1 .0 LTE Done 5 of 5 (3) Let V be the vector space of polynomials of degree at most n. Are the following subspaces? Prove or disprove. (a) All polynomials of the form f(x) = ar", where a is a...

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...

Problem 7.1. The trace of a square matrix A, written tr(A), is the sum of its diagonal entries. For example, N tr = 1+5+9 = 15. In the following, Y denotes an arbitrary n x m matrix. (a) Verify that...

please help me in this 3 questions 4. (5 pts.) Let V be a vector space. Let 7: V - V be a linear transformation such that To T = 1. Show that {v, T(v) } is linearly dependent if and only if T(v) =...

4.6.A Let A be a m x 72 matrix with m 2 n. What is the largest possible dimension of (a) the null space of A? (b) the column space of A? (c) the row space of A? 4.6.B Let A be a m x n matrix with m S...

Juicy Wiggle Limited, a company involved in the manufacture of energy juices, purchased new manufacturing machine on the 1 March 2017 for an amount of R275 000. The machine had a useful life of 5...

Use the below information to value the debt in a levered company with annual perpetual cash flows from assets that grow. The next cash flow will be generated in one year from now. Data on a Levered...

The strength of a patient's reaction to a dose of x milligrams of a certain drug is R ( x ) = 8 x 1 6 + 0 . 5 x for 0 x 1 6 0 . The derivative R ' ( x ) is called the sensitivity to the drug. Find R...

You are the project manager of the DFK Project and youre reviewing your project network diagram. You are considering changing the relationship of some of the project activities to reduce the duration...

For each pair of function in Exercise f(x) = 2x + 3, g(x) = 3 - 5x (a) Find the domain of f, g, f + g, f - g, fg, f f, f/g, and g/f. (b) Find (f + g) (x), (f- g) (x), (fg) (x), (ff) (x), (f/g) (x),...

Consider the functions F and G as shown in the graph below. (a) Find the domain of F, the domain of G, and the domain of F + G. (b) Find the domain of F - G, FG, and F/G. (c) Find the domain of G/F....

A small biotechnology research corporation has been experiencing losses for the first three years of its existence, and thus has a negative balance in retained earnings. The corporations stock price,...

Fabulous Fabricators needs to decide how to allocate space in its production facility this year. It is considering the following contracts: B a. What are the profitability indexes of the projects? b....

4 . Shares of ABC Ltd are currently traded at Rs . 1 7 7 0 . A call option with a strike price of 1 7 5 0 and expiring in 9 0 days is traded at Rs . 6 0 while a put with same strike price and...