prelimsolutions1 - Prelim I Solutions 1. We are given a...

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Prelim I Solutions 1. We are given a system of 4 linear equations in 7 unknowns in the form A~x = ~ b , and told that the augmented matrix A ~ b has reduced row echelon form 1 2 0 0 -2 0 4 3 0 0 0 1 3 0 1 2 0 0 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 1. Find all solutions to A~x = ~ b . Solution: x 1 + 2 x 2 - 2 x 5 + 4 x 7 = 3 x 4 + 3 x 5 + x 7 = 2 x 6 - x 7 = 1 Let x 2 = a,x 3 = b,x 5 = c,x 7 = d x 1 = 3 - 2 a + 2 c - 4 d x 4 = 2 - 3 c - d x 6 = 1 + d 2. Let T : R p R q be the linear transformation given by T ( ~x ) = A~x . Determine p and q . Solution: Since the matrix of the transformation is a 4 × 7 p = 7 q = 4 3. Determine rank( A ) and the dimension of image( T ). Solution: Rank of A is the number of leading 1’s in the matrix which is 3, same as the dimension of the image space. 4. Is it true that ~ b is an element of image( T )? Why or why not? Solution: Yes, since ~ b is the output when the matrix of the transformation acts on any solution of part(a), ~ b is in the image of T.
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5. Determine the dimension and a basis for the kernel of T . Solution: By rank nullity, dimension of the basis would be 4. Solving for the system x 1 + 2 x 2 - 2 x 5 + 4 x 7 = 0 x 4 + 3 x 5 + x 7 = 0 x 6 - x 7 = 0 ~x = a - 2 1 0 0 0 0 0 + b 0 0 1 0 0 0 0 + c 2 0 0 - 3 1 0 0 + d -
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prelimsolutions1 - Prelim I Solutions 1. We are given a...

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