# Math 237 Quiz 10

1. Prove the Solvability Criterion for Systems of Linear Equations.

** Solution : **The Solvability Criterion for Systems of Linear Equations

says that any system of linear equations AX = B is solvable if and only

if its vector of constants B belongs to the column space of its coefficient

matrix A (see Theorem 1 on p. 73 of the textbook). Suppose that A_{1},

A_{2}, ..., A_{n} are the columns of A (in this order). Then

where
(see p. 72 of the textbook). So if AX =

B is solvable , then B is contained in the span of the column vectors of A

(= column space of A) and conversely, if B is contained in the column

space of A, then
is a solution of AX = B and

therefore AX = B is solvable .

2. Prove the Translation Theorem for Systems of Linear Equations .

Solution : The Translation Theorem for Systems of Linear Equations

says that S_{B} = T + S_{0} for any particular solution T of the system

AX = B of linear equations . (Here S_{B} denotes the set of solutions of

the system AX = B and T + S_{0} := {T + Z | Z ∈S_{0}} = {T + Z |

AZ = 0}.) Let us begin by proving the inclusion
(see also the

footnote on p. 102 of the textbook). So suppose that S is any solution

of AX = B. Then A(S − T) = AS − AT = B − B = 0 shows

that S − T ∈S_{0} or equivalently S ∈T + S_{0}. Now let us prove the

reverse inclusion So suppose that S = T + Z with Z
∈S_{0}. Then

AS = A(T + Z) = AT + AZ = B + 0 = B shows that S is a solution

of AX = B.

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