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Section 3.2 The Kernel

Definition 3.2.1.

Given a linear transformation \(T:\mathbb{L}_1 \longrightarrow \mathbb{L}_2\) the set of points in \(\mathbb{L}_1\) that \(T\) sends to \(\mathbf{0} \in \mathbb{L}_2\) is called the kernel of \(T\) and written:

\begin{equation*} ker(T)= \{ \mathbf{P} \in \mathbb{L}_1 \ |\ T(\mathbf{P}) = \mathbf{0} \} \end{equation*}

Describe the kernel of \(T:\mathbb{R}^3 \longrightarrow \mathbb{R}^3\) if T is the linear transformation that

  1. rotates all points around the z-axis by 90\(^\circ\)

  2. projects all points perpendicularly onto the xy-plane

  3. projects all points horizontally onto the z-axis

  4. reflects all points across the xy-plane

  5. moves all points straight out to twice their distance from origin

What is the kernel of \(T:C[0,1] \longrightarrow C[0,1]\) if \(T\) is the linear transformation defined by \(T(f) = f'\text{.}\)

For \(T:\mathbb{R}^2 \longrightarrow \mathbb{R}^3\) find \(ker(T)\) and \(dim(ker(T))\) if T is the linear transformation defined by

  1. \(\displaystyle T(x,y) = (0, 0, 0)\)

  2. \(\displaystyle T(x,y) = (0, 0 ,x+y)\)

  3. \(\displaystyle T(x,y) = (x,y,0)\)

  4. \(\displaystyle T(x,y) = (x, y, x+y)\)