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.

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.

answer

d. 10

explanation

the distance from point y to wx is the height of the triangle.

using pythagoras theorem,

group similar terms to get,

this implies that,

take positive square root of both sides to get,

this implies that,

hi there!

the answer will be d: 2