Question

In Exercise 23 and 24, mark each statement True or False. Justify each answer.

24.

a. If \(x\) is a nontrivial solution of \(Ax = 0\), then every entry in \({\mathop{\rm x}\nolimits} \) is nonzero.

b. The equation \(x = {x_2}{\mathop{\rm u}\nolimits} + {x_3}{\mathop{\rm v}\nolimits} \) , with \({x_2}\) and \({x_3}\) free (and neither \({\mathop{\rm u}\nolimits} \) nor \(v\) a multiple of the other), describes a plane through the origin.

c. The equation \(Ax = b\) is homogeneous if the zero vector is a solution.

d. The effect of adding \({\mathop{\rm p}\nolimits} \) to a vector is to move the vector in a direction parallel to \({\mathop{\rm p}\nolimits} \).

e. The solution set of \(Ax = b\) is obtained by translating the solution set of \(Ax = 0\).

Short answer

a. The given statement isfalse.

b. The given statement istrue.

c. The given statement istrue.

d. The given statement istrue.

e. The given statement isfalse.

Step-by-step solution

(a) Identify whether the statement is true or false

a.

Anontrivial solution \(x\) can have some zero entries until and unless all of its entries are zero.

Thus, the given statement (a) is false.

(b) Identify whether the statement is true or false

b.

As it is known that neither \({\mathop{\rm u}\nolimits} \) nor \({\mathop{\rm v}\nolimits} \) is a scalar multiple of the other, a solution set is a plane through the origin.

Thus, the given statement (b) is true.

(c) Identify whether the statement is true or false

c.

A homogeneous linear system \(Ax = 0\) always has at least one solution \(x = 0\). Zero vector is a solution because \(b = Ax = A0 = 0\) .

Thus, the given statement (c) is true.

(d) Identify whether the statement is true or false

d.

It is known that the effect of adding \({\mathop{\rm p}\nolimits} \) to \({\mathop{\rm v}\nolimits} \) moves \({\mathop{\rm v}\nolimits} \) in a direction parallel to the line through 0 and \({\mathop{\rm v}\nolimits} \).

Thus, the given statement (d) is true.

(e) Identify whether the statement is true or false

e.

The solution set is obtained by translating the solution set of \(Ax = 0\). The statement holds only if the solution set \({\mathop{\rm Ax}\nolimits} = 0\) is nonempty.

Thus, the given statement (e) is false.