A First Course in Linear Algebra · Problem ILT-C40

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · injective-linear-transformations

Problem

Show that the linear transformation

R

is not injective by finding two different elements of the domain,

\mathbf{x}

and

\mathbf{y}

, such that

R(\mathbf{x}) = R(\mathbf{y})

. (

S_{22}

is the vector space of symmetric

2 \times 2

matrices.)

R: S_{22} \to P_1 \qquad R\begin{pmatrix} \begin{bmatrix} a & b \\ b & c \end{bmatrix} \end{pmatrix} = (2a - b + c) + (a + b + 2c)x