Problem 1

Let and be bases for vector spaces and , respectively. Let be a linear transformation with the property that

Find the matrix for relative to and .

Problem 2

Let and be bases for vector spaces and , respectively. Let be a linear transformation with the property that

Find the matrix for relative to and .

Problem 3

Let be the standard basis for , be a basis for a vector space , and be a linear transformation with the property that

(a). Compute , , and

(b). Compute , , and

(c). Find the Matrix for Relative to and

Problem 4

Let be a basis for a vector space and be a linear transformation with the property that

Find the matrix for relative to and the standard basis for .

Problem 5

Let be the transformation that maps a polynomial into the polynomial .

(a). Find the Image of

(b). Show that is a Linear Transformation

(c). Find the Matrix for Relative to the Bases and

Problem 6

Let be the transformation that maps a polynomial into the polynomial .

(a). Find the Image of

(b). Show that is a Linear Transformation

(c). Find the Matrix for Relative to the Bases and

Problem 7

Assume the mapping defined by

is linear. Find the matrix representation of relative to the basis .

Problem 8

Let be a basis for a vector space . Find when is a linear transformation from to whose matrix relative to is

Problem 9

Define by .

(a). Find the Image under of

(b). Show that is a Linear Transformation

(c). Find the Matrix for Relative to the Basis for and the Standard Basis for

Problem 10

Define by .

(a). Show that is a Linear Transformation

(b). Find the Matrix for Relative to the Basis for and the Standard Basis for

Problem 11

Find the -matrix for the following transformations , when .

(a).

(b).