In Exercises 23 and 24, mark each statement True or False. Justify each answer.
ID: 3027916 • Letter: I
Question
In Exercises 23 and 24, mark each statement True or False. Justify each answer. A homogeneous equation is always consistent. The equation Ax = 0 gives an explicit description of its solution set. The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable The equation x = p + t v describes a line through v parallel to p The solution set of Ax = b is the set of all vectors of the form w = p + vh, where vh is any solution of the equation Ax = 0Explanation / Answer
a) TRUE.
Solving a homogeneous equation Ax = 0 is equivalent to finding the kernel of the linear transformation T corresponding to the matrix A.
Since the kernel of any linear trasformation is always well-defined, a homogeneous system is always consistent.
b) TRUE.
Since the solution set is but the kernel of the transformation, it can be explicitly found out.. This can be achieved by first reducing the matrix to its row echelon form and then solving backwards.
c) FALSE.
On the contrary, the equation has the trivial solution if and only if the system of equations DOES NOT have a free variable.
d) FALSE.
The mentioned equation describes the line passing through p and parallel to v.
e) TRUE.
If p is solution of Ax = b and v is a solution of Ax = 0 then for any real number h and for w = p + vh we have,
A(w) = A(p + vh) = A(p) + A(vh) = b + hA(v) = b + 0 = b.