QuestionHow many rows and columns must matrix $A$ have to map $\mathbb{R}^{8}$ to $\mathbb{R}^{9}$ using $T(x)=A x$ ? A. 9 rows, 9 columns B. 9 rows, 8 columns C. 8 rows, 8 columns D. 8 rows, 9 columns

Studdy Solution

STEP 1

Assumptions1. We are given a linear transformation $(x)=Ax$ from $\mathbb{R}^{8}$ to $\mathbb{R}^{9}$ . . We need to find the dimensions of the matrix $A$ that defines this transformation.

STEP 2

In a linear transformation defined by a matrix multiplication, the number of columns in the matrix must equal the dimension of the input space. This is because each column of the matrix corresponds to a basis vector in the input space.
$\text{Number of columns in } A = \text{Dimension of input space}$

STEP 3

Substitute the given dimension of the input space into the equation.
$\text{Number of columns in } A =8$

STEP 4

Similarly, the number of rows in the matrix must equal the dimension of the output space. This is because each row of the matrix corresponds to a basis vector in the output space.
$\text{Number of rows in } A = \text{Dimension of output space}$

STEP 5

Substitute the given dimension of the output space into the equation.
$\text{Number of rows in } A =9$ So, the matrix $A$ must have9 rows and8 columns. Therefore, the correct answer is B. The matrix A must have9 rows and8 columns.

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