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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

Expert verified
  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

01

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.

02

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.

03

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.

04

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.

05

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.

06

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.

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Most popular questions from this chapter

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)

What would you have to know about the solution set of a homogenous system of 18 linear equations 20 variables in order to understand that every associated nonhomogenous equation has a solution? Discuss.

Suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. Is it possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other? Discuss.

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations? If so, how many? Discuss.

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