QuestionA denotes an $m \times n$ matrix. Determine whether the statement is true or false. Justify your answer. The column space of $A, \operatorname{Col} A$ , is the set of all solutions of $A x=b$ .
Choose the correct answer below. A. The statement is true. The column space of $A$ is $\operatorname{Col} A=\left\{b: b=A x\right.$ for some $\mathbf{x}$ in $\left.\mathbb{R}^{n}\right\}$ . B. The statement is false. The column space of $A$ is the set of all solutions of $A \mathbf{x}=\mathbf{0}$ . C. The statement is false. The column space of $A$ is $\operatorname{Col} A=\left\{b: b=A x\right.$ for some $\mathbf{x}$ in $\left.\mathbb{R}^{n}\right\}$ . D. The statement is true. The column space of $A$ is the set of all solutions of $A \mathbf{x}=\mathbf{0}$ .

Studdy Solution

STEP 1

1. We are given a matrix $A$ of size $m \times n$ .
2. We need to determine the nature of the column space of $A$ .
3. The column space of $A$ , denoted as $\operatorname{Col} A$ , is defined as the set of all linear combinations of the columns of $A$ .

STEP 2

1. Define the column space of a matrix.
2. Determine the relationship between the column space and the equation $A \mathbf{x} = \mathbf{b}$ .
3. Choose the correct answer based on the definition and relationship.

STEP 3

Define the column space of a matrix $A$ :
The column space of $A$ , denoted as $\operatorname{Col} A$ , is the set of all vectors that can be expressed as linear combinations of the columns of $A$ . Mathematically, it is represented as:
$\operatorname{Col} A = \{ \mathbf{b} : \mathbf{b} = A \mathbf{x} \text{ for some } \mathbf{x} \in \mathbb{R}^n \}$

STEP 4

Determine the relationship between the column space and the equation $A \mathbf{x} = \mathbf{b}$ :
The column space $\operatorname{Col} A$ represents all possible vectors $\mathbf{b}$ that can be obtained by multiplying $A$ with some vector $\mathbf{x}$ . Therefore, $\operatorname{Col} A$ is the set of all vectors $\mathbf{b}$ for which the equation $A \mathbf{x} = \mathbf{b}$ has at least one solution.

STEP 5

Choose the correct answer based on the definition and relationship:
Option A is correct because it states that the column space of $A$ is the set of all vectors $\mathbf{b}$ such that $\mathbf{b} = A \mathbf{x}$ for some $\mathbf{x}$ in $\mathbb{R}^n$ , which aligns with our definition.
Therefore, the correct answer is:
A. The statement is true. The column space of $A$ is $\operatorname{Col} A = \{ \mathbf{b} : \mathbf{b} = A \mathbf{x} \text{ for some } \mathbf{x} \in \mathbb{R}^n \}$ .

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