Matrix Representation of a Linear Transformation

To find the matrix representation of a linear transformation, we need to determine how the transformation behaves with respect to the basis vectors. In this case, we are given the linear transformation T: ℝ → ℝ² defined by T(x) = (3x, 5x), where x is a scalar.

Step 1: Define the Basis Vectors
The usual basis for ℝ is {1}, and the usual basis for ℝ² is {(1,0), (0,1)}. Therefore, the basis vectors we will consider are 1 and (1,0).

Step 2: Apply the Transformation to the Basis Vectors
To find the matrix representation, we apply the transformation T to each basis vector.

T(1) = (3(1), 5(1)) = (3, 5)
T((1,0)) = (3(1), 5(1)) = (3, 5)

Step 3: Write the Results as Column Vectors
The results obtained in Step 2 will be written as column vectors.

(3, 5) = 3(1, 0) + 5(0, 1)
(3, 5) = 3(1, 0) + 5(0, 1)

Step 4: Construct the Matrix
The matrix representation of the linear transformation is obtained by arranging the column vectors from Step 3 as the columns of a matrix.

[ 3 0 ]
[ 5 1 ]

Explanation
The matrix [3 0; 5 1] represents the linear transformation T: ℝ → ℝ² defined by T(x) = (3x, 5x). This means that for any scalar x, the transformation T will multiply the input by 3 to obtain the first component of the output and by 5 to obtain the second component of the output.

By applying the transformation to the basis vectors, we see that T(1) = (3, 5) and T((1,0)) = (3, 5). This implies that the transformation maps the basis vectors 1 and (1,0) to the column vectors (3, 5) and (3, 5) respectively.

The matrix representation [3 0; 5 1] can be used to perform the linear transformation on any vector in ℝ. To do this, we simply multiply the matrix by the vector using matrix multiplication. This matrix multiplication will yield the corresponding vector in ℝ².

In conclusion, the matrix representation of the linear transformation T: ℝ → ℝ² defined by T(x) = (3x, 5x) is [3 0; 5 1].