Kunze Linear Algebra · Problem 3.4.2

Linear Transformations

Mathematics — Methods · matrix-of-transformation-r3-to-r2

Problem

Let

T

be the linear transformation from

R^3

into

R^2

defined by

T(x_1, x_2, x_3) = (x_1 + x_2,\; 2x_3 - x_1).

(a) If

\mathfrak{B}

is the standard ordered basis for

R^3

and

\mathfrak{B}'

is the standard ordered basis for

R^2

, what is the matrix of

T

relative to the pair

\mathfrak{B}

\mathfrak{B}'

? (b) If

\mathfrak{B} = \{\alpha_1, \alpha_2, \alpha_3\}

and

\mathfrak{B}' = \{\beta_1, \beta_2\}

, where

\alpha_1 = (1, 0, -1), \quad \alpha_2 = (1, 1, 1), \quad \alpha_3 = (1, 0, 0), \quad \beta_1 = (0, 1), \quad \beta_2 = (1, 0)

what is the matrix of

T

relative to the pair

\mathfrak{B}

\mathfrak{B}'