<exercise>
<statement>
<p>Consider the following maps of polynomials <m>S:\mathcal{P}\rightarrow\mathcal{P}</m> and <m>T:\mathcal{P}\rightarrow\mathcal{P}</m> defined by
<me>
S(<xsl:value-of select="f_letter"/>(x))=
<xsl:choose>
<xsl:when test="swapped">
<xsl:value-of select="nonlinear_trans"/>
</xsl:when>
<xsl:otherwise>
<xsl:value-of select="linear_trans"/>
</xsl:otherwise>
</xsl:choose>
\hspace{1em} \text{and} \hspace{1em} T(<xsl:value-of select="f_letter"/>(x))=
<xsl:choose>
<xsl:when test="swapped">
<xsl:value-of select="linear_trans"/>
</xsl:when>
<xsl:otherwise>
<xsl:value-of select="nonlinear_trans"/>
</xsl:otherwise>
</xsl:choose>
</me>
Explain why one these maps is a linear transformation and why the other map is not.
</p>
</statement>
<answer>
<xsl:choose>
<xsl:when test="swapped">
<p><m>S</m> is not linear and <m>T</m> is linear.</p>
</xsl:when>
<xsl:otherwise>
<p><m>S</m> is linear and <m>T</m> is not linear.</p>
</xsl:otherwise>
</xsl:choose>
</answer>
</exercise>
