Question

Write a short paragraph to answer each of the following questions about solutions of systems of linear equations. You should not perform any calculations, but instead base your explanations on the appropriate properties.

[1] One solution of the homogeneous linear system
x + 2y + z + 3w = 0
x - y + + w = 0
y - z + 2w = 0
is x=-2, y=-1, z=1, ans w=1. Explain why x=4, y=2, z=-2, and w=-2 must also be a solution. Do not perform any row operations.

[2] The vectors x1 and x2 are solutions of the homogeneous linear system Ax=0. Explain why the vector 2x1-3x2 must also be a solution.
[3] Consider the two systems represented by the augmented matrices.
[ 1 1 -5 3; 1 0 -2 1; 2 -1 -1 0] [ 1 1 -5 -9; 1 0 -2 -3; 2 -1 -1 0]
If the first system is known to be consistent, explain why the second system is also consistent. Do not perform any row operations.

[4] The vectors x1 and x2 are solutions of the linear system Ax=b. Is the vector 2x1-3x2 also a solution? Why or why not?

[5] The linear systems Ax = b1 + b2 necessarily consistent? Why or why not?

[6] Consider the linear system Ax = b. If the rank of A equals the rank of the augmented matrix for the system, explain why the system must be consistent. Contrast this to the case in which the rank of A is less than the rank of the augmented matrix.

Answer 1

Given  : One solution of the homogeneous linear system

x + 2y + z + 3w = 0

x - y + + w = 0

y - z + 2w = 0

is x=-2, y=-1, z=1, and w=1.

To Find :   Explain why x=4, y=2, z=-2, and w=-2

must also be a solution.

Solution:

x + 2y + z + 3w = 0

x - y + + w = 0

y - z + 2w = 0

multiplying by a constant  k

=> kx + 2(ky) + z + 3(kw) = 0

kx - ky + + kw = 0

ky - kz + 2kw = 0

Hence   x=-2, y=-1, z=1, and w=1.

=> kx=-2, ky=-1, kz=1, and kw=1

substituting k = -1/2

=> x =  4 , y = 2  , z  = - 2  and  w = -2

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