Question

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

23.

a. A homogeneous equation is always consistent.

b. The equation \(Ax = 0\) gives an explicit description of its solution set.

c. The homogeneous equation \(Ax = 0\) has the trivial solution if and only if the equation has at least one free variable.

d. The equation \(x = p + tv\) describes a line through \({\mathop{\rm v}\nolimits} \) parallel to \(p\).

e. The solution set of \(Ax = b\) is the set of all vectors of the form \({\mathop{\rm w}\nolimits} = p + {v_k}\), where \({v_k}\) is any solution of the equation \(Ax = 0\).

Short answer

a. The given statement istrue.

b. The given statement isfalse.

c. The given statement isfalse.

d. The given statement isfalse.

e. The given statement isfalse.

Step-by-step solution

(a) Identify whether the statement is true or false

a.

\(A\)is a \(m \times n\) matrix and \(0\) is the zero vector in \({\mathbb{R}^m}\). Such a homogeneous system \(Ax = 0\) always has at least one solution \(x = 0\). This zero solution is called the trivial solution.

Thus, the given statement (a) is true.

(b) Identify whether the statement is true or false

b.

It is known that the equation \(Ax = 0\) provides an implicit description of a solution set.

Thus, the given statement (b) is false.

(c) Identify whether the statement is true or false

c.

The homogeneous equation \(Ax = 0\) has anontrivial solution if and only if the equation has at least one free variable.

Thus, the given statement (c) is false.

(d) Identify whether the statement is true or false

d.

The equation of the line passing through \({\mathop{\rm p}\nolimits} \) and parallel to \({\mathop{\rm v}\nolimits} \)is written as \(x = {\mathop{\rm p}\nolimits} + t{\mathop{\rm v}\nolimits} \).

Thus, the given statement (d) is false.

(e) Identify whether the statement is true or false

e.

It is known that the solution set is empty. The statement is true if and only if there is a vector \({\mathop{\rm p}\nolimits} \) such that \(A{\mathop{\rm p}\nolimits} = b\).

Thus, the given statement (e) is false.