FinalTestReviewSpring2011

# B find the most general solution of a w for this

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: value of number c vector w = ￿ belongs 1 2 0 2 5 4 −2 c to the column space of A? (b) Find the most general solution of A￿ = w for this value of c. x￿ (c) Without further calculation, identify a basis for a row space of A (i.e., for C (AT )). ( pts) Solution: (a) Vector w belongs to C (A) precisely when system A￿ = w has a ￿ x ￿ solution. Thus, we need to solve it. We will do it by reducing augmented matrix of this system: 11 0 2 1 1 0 2 −￿2 r 1 0 −2 4 0 2 0 0 1 4 −4 × 1/2 1 2 −2 2 −2 → → 1 2 0 0 0 2 0 −￿1 r 1 2 −2 −￿2 r 0 0 5 4 −2 c −5￿1 r 0 −1 −2 c − 10 +￿2 r 00 0 c − 12 For this system to have the solution we need to have c − 12 = 0, that is, c = 12. (b) We already have done all reduction work for this problem in part (a). The free variable is x3 , since the third column is the only non-pivot column. From the ﬁrst reduced equation x1 − 2x3 = 4, we get x1 = 4 + 2x3 . The second equation x2 + x3 =−2 2 gives x2 = − − 2x. 2 3 x1 4 −2...
View Full Document

Ask a homework question - tutors are online