Math 217 Winter 2009 - Group Homework 1 Solutions
1. Consider a general system of linear equations
a11x1+a12x2+· · · +a1nxn=b1
a21x1+a22x2+· · · +a2nxn=b2
.
.
.
am1x1+am2x2+· · · +amn xn=bm
This system has mequations in nvariables.
Prove that any of the elementary row operations applied to this system
results in an equivalent system.
Solution: We must show that, for each of the three basic row opera-
tions, when the operation is applied to the above system, the resulting
system has the same solution set as the above system. First, for the
operation of interchanging two rows, the resulting system of equations
literally consists of the same set of equations, just in a different order.
It follows immediately that they have the same solution sets.
Let cbe a nonzero real number, let kbe an integer with 1 ≤k≤m,
and consider the operation Rk−→ c·Rkof multiplying row kby c. If
~x =
x1
.
.
.
xn
is a solution of the original system, then multiplying the kth equation
by cpreserves the equality (since the same thing has been done to both
sides of that equation). It follows that ~x is a solution to the transformed
system. Conversely, if ~x is a solution to the transformed system, then
multiplying the kth equation by c−1(this exists since c6= 0) again
preserves the equality and results in the kth equation in the original
system, so ~x is a solution to the original system. Thus we have shown
that ~x is a solution to the original system if and only if ~x is a solution
to the transformed system, which means that these solutions have the
same solution set, and are therefore equivalent by definition.
Lastly, let k, l be distinct integers with 1 ≤l, k ≤m, let c∈Rbe
any real number, and consider the operation Rk−→ Rk+c·Rlof
