Determine whether the following statements are true or false. Justify briefly.

Assume the solution set of a homogeneous linear system with coeficient matrix A is a line then the solution set of a consistent system with coecient matrix A is also a line.

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Statement is false.

For example

consider following Homogeneous system of equations

8x+12y=0

2x+3y=0

Coefficient matrix

A=`[[8,12],[2,3]]`

The solution of this system is line y=(-2/3)x

Consider following homogeneous system of equation

x+y=0

3x+4y=0

It has only trivial solution

x=y=0

This solution not represent line but a point (0,0)

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