Matrix Representation of a Linear Transformation:

To find the matrix representation of a linear transformation, we need to determine how the transformation T maps the standard basis vectors in the domain to vectors in the codomain.

Linear Transformation T: R → R²:

The linear transformation T is defined as T(x) = (3x, 5x), where x is a real number. This means that for any real number x in the domain, T maps it to a vector in R² with components (3x, 5x).

Standard Basis for R:

The standard basis for R is {1}, which means that any real number x in the domain can be written as a linear combination of the basis vector 1.

Matrix Representation:

To find the matrix representation of T relative to the usual basis for R → R², we need to determine how T maps the basis vector 1 to vectors in R².

Mapping of Basis Vector 1:

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

The vector (3, 5) in R² represents the image of the vector 1 under the linear transformation T.

Matrix Representation:

The matrix representation of T relative to the usual basis is a 2x1 matrix, where each column represents the components of the image of the corresponding basis vector.

Since T(1) = (3, 5), the matrix representation of T is:

[3]
[5]

This 2x1 matrix represents how the linear transformation T maps the basis vector 1 to the vector (3, 5) in R².

Explanation:

The linear transformation T maps real numbers x to vectors (3x, 5x) in R². By determining how T maps the basis vector 1, we can find the matrix representation of T relative to the usual basis. In this case, T(1) = (3, 5), so the matrix representation of T is [3, 5]. This matrix represents how T maps the basis vector 1 to the vector (3, 5) in R².