Question

Linear Algebra Problem: Determine if the given statements are True or False. Justify your answers (i.e., give a proof for true and a counterexample for false).
(a) If a linear system has more equations than variables, then it must be inconsistent.


(b) If a linear system has free variables, then it must have infinitely many solutions.


(c) Every linear system with the same number of equations and variables has a unique solution.


(d) In every linear system with 2 equations and 3 variables x1,x2,x3, the last variable x3 is a free variable.


(e) If x1 = x2 = ··· = xn = 0 is a solution to a linear system, then the system must be homogeneous.

Answer 1

Answer:

a) False

b)True

c)False

d)True

e)True

Step-by-step explanation:

a) False

No it is not true for all cases

Ex

  2 x = 5 ------1

3 x =7.5 ------2

So here equation is 2 but we have a value of x=2.5.

b)True

If a linear system has free variables, then it must have infinitely many solutions.

c)False

Every linear system with the same number of equations and variables has a unique solution.This is not true for all cases.

d)True

In every linear system with 2 equations and 3 variables x1,x2,x3, the last variable x3 is a free variable.

e)True

If x1 = x2 = ··· = xn = 0 is a solution to a linear system, then the system must be homogeneous.