True or False: A system of linear equations with fewer equations than variables always...?

Question

...has at least one solution.
If false, please explain.
Thank you.

Answer 1

This is false.

For example, take the following two equations in three variables:

x + y + z = 1
x + y + z = 2

Obviously, no point (x, y, z) can satisfy both these equations, because the same sum can't equal 1 and 2 at the same time, no matter which point (x, y, z) you pick.

Something that IS true is "If a system of linear equations with fewer equations than variables has at least one solution, then it always has infinitely many solutions."

Answer 2

False. Here are three valid linear equations:

X = 2 ,
X = 3,
Y + Z + W = 4.

The two first equations are as simple as possible, and have no solution. There are three equations, four variables, and yet no solution.

Answer 3

a system of linear equations with the same number of equations as variables can be solved for all variables.

with less equations than variables, the best you can hope for is to solve for the same number of variables as you have equations.

Answer 4

F..A..L..S..E.
The answer hinges upon the definition of a "basis" for "n-space" for n variables. The basis must contain AT LEAST as many vectors in n-space as there are dimensions.