Need help with these questions! Please Help! 1. [10 pointsl Let A=12 5 1-1 i) Fi

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Need help with these questions! Please Help!

1. [10 pointsl Let A=12 5 1-1 i) Find the columns of A which constitute a basis of its column space;ii Write the nonbasis columns of A into linear combinations of the basis columns; i) Find the nullspace of A and determine a basis. iv) Find the dimensions of the row space and column space. 3. [10 points] i) Show that N(A) = N(EA) if E is invertible. ii) Construct an example of a 2 x 2 matrix A and an elementary matrix E such that C(A-C(EA). 4. [10 points] Let A be i) Find all possible vectors x such that x E R(A)nN(A), where R(A) denotes the row space of A. ii) Let m> n. Show that AAT is not invertible. 5. [10 points] Let A = | 4 5 6|. i) Find N(A) and split x = (1, 1, 1) into x = zr+Zn with In EN(A) and xrE R(A); Find a basis of the orthogonal complement of N(A) [10 points] Let Rn be a nonzero vector. Show that ) V = {xER'. : x . = 0} is a subspace of R": ii) dim V = n-1. 7. [10 points] Let A E Mnn be such that A2A and ATA. Show that i) N(A) = C(I-A); ii) C(A) C(1-A), where C(A) denotes the column space of A; iii) every x Rn can be uniquely decomposed as x = xc+x, where xeE C(A) and Xi C(1-A).

Explanation / Answer

(i) For determining the columns of A which columns constitute a basis for Col(A), we will reduce A to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add -3 times the 1st row to the 3rd row

Multiply the 2nd row by -1/3

Add 5 times the 2nd row to the 3rd row

Multiply the 3rd row by ¾

Add -17/3 times the 3rd row to the 2nd row

              Add -9 times the 3rd row to the 1st row

Add -4 times the 2nd row to the 1st row

Then the RREF of A is

1

0

0

193/4

0

1

0

-81/4

0

0

1

15/4

Now it is apparent that the first 3 columns of A are linearly independent and constitute Col(A). Thus, a basis for Col(A) is {(1,2,3)T,(4,5,7)T,(9,1,0)T}.

(ii) Further, (1,-1,3)T = (193/4)(1,2,3)T-(85/4)(4,5,7)T+(15/4)(9,1,0)T.

(iii) The Null space of A is the set of solutions of the equation AX = 0. If X = (x,y,z,w)T, then, in view of the RREF of A, this equation is equivalent to x = 193/4, y = -81/4 and z = 15/4. Further, w is arbitrary. Then X = (193/4, -81/4,15/4,w)T = ¼(193,-81,15,0)T+w(0,0,0,1)T. The null space of A is                                         span{(193,-81,15,0)T,(0,0,0,1)T}. A basis for null(A) is {(193,-81,15,0)T,(0,0,0,1)T}.

(iv) The dimension of the row space , being equal to the number of non-zero rows in the RREF of A, is 3. The dimension of Col(A) is same as dimRow(A) , i.e. 3.

Please post the remaining questions again, one at a time.

              Add -9 times the 3rd row to the 1st row

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