QuestionWhat dimensions must matrix $A$ have to map $\mathbb{R}^{6}$ to $\mathbb{R}^{7}$ using $T(\mathbf{x})=A \mathbf{x}$ ? A. 6 rows, 7 columns B. 6 rows, 6 columns C. 7 rows, 6 columns D. 7 rows, 7 columns

Studdy Solution

STEP 1

Assumptions1. The mapping is defined from $\mathbb{R}^{6}$ to $\mathbb{R}^{7}$ . . The mapping is defined by the rule $(\mathbf{x})=A \mathbf{x}$ , where $A$ is a matrix and $\mathbf{x}$ is a vector in $\mathbb{R}^{6}$ .
3. We are looking for the number of rows and columns that the matrix $A$ must have.

STEP 2

In order for the matrix multiplication $A \mathbf{x}$ to be defined, the number of columns in $A$ must be equal to the number of rows in $\mathbf{x}$ .

STEP 3

Since $\mathbf{x}$ is a vector in $\mathbb{R}^{6}$ , it has6 rows. Therefore, the matrix $A$ must have6 columns.

STEP 4

The result of the matrix multiplication $A \mathbf{x}$ is a vector in $\mathbb{R}^{7}$ . This means that the result has7 rows.

STEP 5

In order for the result of the matrix multiplication to have7 rows, the matrix $A$ must also have7 rows.

STEP 6

Therefore, the matrix $A$ must have rows and6 columns.
The correct answer is C. The matrix A must have rows and6 columns.

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QuestionWhat dimensions must matrix $A$ have to map $\mathbb{R}^{6}$ to $\mathbb{R}^{7}$ using $T(\mathbf{x})=A \mathbf{x}$ ? A. 6 rows, 7 columns B. 6 rows, 6 columns C. 7 rows, 6 columns D. 7 rows, 7 columns

Studdy Solution

STEP 1

STEP 2

In order for the matrix multiplication $A \mathbf{x}$ to be defined, the number of columns in $A$ must be equal to the number of rows in $\mathbf{x}$ .

STEP 3

Since $\mathbf{x}$ is a vector in $\mathbb{R}^{6}$ , it has6 rows. Therefore, the matrix $A$ must have6 columns.

STEP 4

The result of the matrix multiplication $A \mathbf{x}$ is a vector in $\mathbb{R}^{7}$ . This means that the result has7 rows.

STEP 5

In order for the result of the matrix multiplication to have7 rows, the matrix $A$ must also have7 rows.

STEP 6

Therefore, the matrix $A$ must have rows and6 columns.
The correct answer is C. The matrix A must have rows and6 columns.

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QuestionWhat dimensions must matrix AAA have to map R6\mathbb{R}^{6}R6 to R7\mathbb{R}^{7}R7 using T(x)=AxT(\mathbf{x})=A \mathbf{x}T(x)=Ax? A. 6 rows, 7 columns B. 6 rows, 6 columns C. 7 rows, 6 columns D. 7 rows, 7 columns

Studdy Solution

STEP 1

STEP 2

In order for the matrix multiplication AxA \mathbf{x}Ax to be defined, the number of columns in AAA must be equal to the number of rows in x\mathbf{x}x.

STEP 3

Since x\mathbf{x}x is a vector in R6\mathbb{R}^{6}R6, it has6 rows. Therefore, the matrix AAA must have6 columns.

STEP 4

The result of the matrix multiplication AxA \mathbf{x}Ax is a vector in R7\mathbb{R}^{7}R7. This means that the result has7 rows.

STEP 5

In order for the result of the matrix multiplication to have7 rows, the matrix AAA must also have7 rows.

STEP 6

Therefore, the matrix AAA must have rows and6 columns.The correct answer is C. The matrix A must have rows and6 columns.

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QuestionWhat dimensions must matrix $A$ have to map $\mathbb{R}^{6}$ to $\mathbb{R}^{7}$ using $T(\mathbf{x})=A \mathbf{x}$ ? A. 6 rows, 7 columns B. 6 rows, 6 columns C. 7 rows, 6 columns D. 7 rows, 7 columns

In order for the matrix multiplication $A \mathbf{x}$ to be defined, the number of columns in $A$ must be equal to the number of rows in $\mathbf{x}$ .

Since $\mathbf{x}$ is a vector in $\mathbb{R}^{6}$ , it has6 rows. Therefore, the matrix $A$ must have6 columns.

The result of the matrix multiplication $A \mathbf{x}$ is a vector in $\mathbb{R}^{7}$ . This means that the result has7 rows.

In order for the result of the matrix multiplication to have7 rows, the matrix $A$ must also have7 rows.

Therefore, the matrix $A$ must have rows and6 columns.
The correct answer is C. The matrix A must have rows and6 columns.