Linear algebra [A|b vector] rref R Now , you are given that the rref of A^T is:

Question

Linear algebra

[A|b vector]

rref

R

Now , you are given that the rref of A^T is: (we don't include b vector in the transpose)

c. Now use the the rref of A^T,, to find another basis for the rowspace of A. Hint : rowspace of A become column of A^T.

d. Express 3rd row of A as linear combination of the basis you found in part (c).

e. Now use the information from the rref of A^T, to find another basis for the columnspace of A. HINT: THE COLUMNS OF A BECOME THE ROWS OF A^T

f. Express 4th column of A as a linear combination of the basis you found in part (e)

3 -5 1 -3 -3 4 3 4 -6 4 -2 -3 2 8 1 -5 -13 -11 -6 18 -19 -2 4 2 4 2 2 0 26 -38 30 -10 -23 48 48

Explanation / Answer

c. Fromthe RREF of AT, it is apparent that a basis for Row(A) , which is same as Col(AT) is { (1,0,0,0,0,0),(0,1,0,0,0,0), (0,0,1,0,0,0)} or, {( (3,-5,1,-3,-3,4),(4,-6,4,-2,-3,2)} provided there have not been any row interchanges in arriving at the RREF..

d. (1,-5,-13,-11,-6,18)= 7 (3,-5,1,-3,-3,4)-5 (4,-6,4,-2,-3,2) provided there have not been any row interchanges in arriving at the RREF.

e. As per the RREF of AT, Col(A) has a basis { (1,0,7,0,8)T,(0,1,-5,0,3)T,(0,0,0,1,5)T }.

f. (-3,-2,-11,4,-10)T =4(3,4,1,-2,26)T +3(-5,-6,-5,4,-38).

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