Question

Question: In Exercises 25 and 26, A denotes a \(m \times n\) matrix. Mark each statement True or False. Justify each answer.

26.

a. A null space is a vector space.

b. The column space of a \(m \times n\) matrix is in \({\mathbb{R}^m}\).

c. Col A is the set of all solutions of \(A{\mathop{\rm x}\nolimits} = b\).

d. Nul A is the kernel of the mapping \({\mathop{\rm x}\nolimits} \mapsto A{\mathop{\rm x}\nolimits} \).

e. The range of a linear transformation is a vector space.

f. The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.

Short answer

1. The given statement is true.
2. The given statement is true.
3. The given statement is false.
4. The given statement is true.
5. The given statement is true.
6. The given statement is true.

Step-by-step solution

Determine whether the given statement is true or false

a)

Theorem 2states that the null spaceof an \(m \times n\) matrix \(A\) is a subspaceof \({\mathbb{R}^n}\). Equivalently, the set of all solutions to a system \(Ax = 0\) of \(m\) homogeneous linear equations in \(n\) unknowns is a subspace of \({\mathbb{R}^n}\).

Thus, statement (a) is true.

Determine whether the given statement is true or false

b)

Theorem 3states that the column spaceof an \(m \times n\) matrix \(A\) is a subspaceof \({\mathbb{R}^m}\).

Thus, statement (b) is true.

Determine whether the given statement is true or false

c)

The column spaceof an \(m \times n\) matrix \(A\) is all of \({\mathbb{R}^m}\) if and only if the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) has a solution for each \({\mathop{\rm b}\nolimits} \) in \({\mathbb{R}^m}\).

Thus, statement (c) is false.

Determine whether the given statement is true or false

d)

The kernel and the rangeof T are just the null space and column space of \(A\).

Thus, statement (d) is true.

Determine whether the given statement is true or false

e)

The rangeof T is the set of all vectors in W of the form \(T\left( {\mathop{\rm x}\nolimits} \right)\) for some \({\mathop{\rm x}\nolimits} \) in \(V\).

Thus, statement (e) is true.

Determine whether the given statement is true or false

f)

The set of all the solutions of a homogeneous linear differential equation turns out to be the kernel of a linear transformation.

Thus, statement (f) is true.