(10 points) (Note: parts (a) and (b) are not related, the Ci in each part are no
ID: 3035830 • Letter: #
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(10 points) (Note: parts (a) and (b) are not related, the Ci in each part are not the same.)
3. (10 points) (Note: parts (a) and (b) are not related, the Ci in each part are not the same.) (a) Let the columns of A be vectors named C1, C2, C3, C4 and C5. Given that Alb1lb2] reduces to the matrix given below which of the bi vectors is in col(A)? For the bi in col(A) express it as a linear combination of the Ci vectors. 1 0 -2 0 7 -2 0 1 0 0 0 0 1 2 0 0 0 0 0 0 (b) If (1234) (243 -1)7 is a basis for null(M), and if b 3C -7C3 where the Ci are b?Explanation / Answer
3. (a) The 1st, 2nd and the 5th columns of A i.e. c1, c2 and c5 are linearly idependent and c2 and c4 are linear combinations of c1 and c2. Further, b1 is not in Col(A) as it has 1 in the last row. Since b2= -2c1 -3c5 , hence b is in Col (A).
(b) Let A =
1
2
2
4
3
3
4
-1
Then the RREF of A is
1
0
0
1
0
0
0
0
We know that the vector b is a linear combination of the columns of a matrix M if and only if the equation Mx = b has at least one solution , say xp . The general solution to Mx = b is given by x = xp + xn, where xp is a particular solution of the equation Mx = b and xn is a generic vector in the nullspace of M. Here, xn is a linear combination of (1,2,3,4)T and (2,4,3,-1)T which is the same as a linear combination of ( 1,0,0,0)T and (0,1,0,0)T. Thus, xn is (a,b,0,0)T, wher a,b are arbitrary real numbers.Then, The general solution to Mx = b is x = xp +(a,b,0,0)T where xp is a particular solution of the equation Mx = b.
1
2
2
4
3
3
4
-1
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