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Section 3.3 Image and Kernel (A3)

Definition 3.3.1.

Let \(T: V \rightarrow W\) be a linear transformation. The kernel of \(T\) is an important subspace of \(V\) defined by

\begin{equation*} \ker T = \left\{ \vec{v} \in V\ \big|\ T(\vec{v})=\vec{z}\right\} \end{equation*}
Figure 3.3.2. The kernel of a linear transformation

Activity 3.3.3.

Let \(T: \IR^2 \rightarrow \IR^3\) be given by

\begin{equation*} T\left(\left[\begin{array}{c}x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x \\ y \\ 0 \end{array}\right] \hspace{3em} \text{with standard matrix } \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right] \end{equation*}

Which of these subspaces of \(\IR^2\) describes \(\ker T\text{,}\) the set of all vectors that transform into \(\vec 0\text{?}\)

  1. \(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ a\end{array}\right]}{a\in\IR}\)

  2. \(\displaystyle \setList{\left[\begin{array}{c}0\\0\end{array}\right]}\)

  3. \(\displaystyle \IR^2=\setBuilder{\left[\begin{array}{c}x \\ y\end{array}\right]}{x,y\in\IR}\)

Activity 3.3.4.

Let \(T: \IR^3 \rightarrow \IR^2\) be given by

\begin{equation*} T\left(\left[\begin{array}{c}x \\ y\\z \end{array}\right] \right) = \left[\begin{array}{c} x \\ y \end{array}\right] \hspace{3em} \text{with standard matrix } \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] \end{equation*}

Which of these subspaces of \(\IR^3\) describes \(\ker T\text{,}\) the set of all vectors that transform into \(\vec 0\text{?}\)

  1. \(\displaystyle \setBuilder{\left[\begin{array}{c}0 \\ 0\\ a\end{array}\right]}{a\in\IR}\)

  2. \(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ a\\ 0\end{array}\right]}{a\in\IR}\)

  3. \(\displaystyle \setList{\left[\begin{array}{c}0\\0\\0\end{array}\right]}\)

  4. \(\displaystyle \IR^3=\setBuilder{\left[\begin{array}{c}x \\ y\\z\end{array}\right]}{x,y,z\in\IR}\)

Activity 3.3.5.

Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by the standard matrix

\begin{equation*} T\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right]\right) = \left[\begin{array}{c} 3x+4y-z \\ x+2y+z \end{array}\right] \end{equation*}

(a)

Set \(T\left(\left[\begin{array}{c}x\\y\\z\end{array}\right]\right) = \left[\begin{array}{c}0\\0\end{array}\right]\) to find a linear system of equations whose solution set is the kernel.

(b)

Use \(\RREF(A)\) to solve this homogeneous system of equations and find a basis for the kernel of \(T\text{.}\)

Activity 3.3.6.

Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by

\begin{equation*} T\left(\left[\begin{array}{c} x \\ y \\ z \\ w \end{array}\right] \right) = \left[\begin{array}{c} 2x+4y+2z-4w \\ -2x-4y+z+w \\ 3x+6y-z-4w\end{array}\right]. \end{equation*}

Find a basis for the kernel of \(T\text{.}\)

Definition 3.3.7.

Let \(T: V \rightarrow W\) be a linear transformation. The image of \(T\) is an important subspace of \(W\) defined by

\begin{equation*} \Im T = \left\{ \vec{w} \in W\ \big|\ \text{there is some }\vec v\in V \text{ with } T(\vec{v})=\vec{w}\right\} \end{equation*}

In the examples below, the left example's image is all of \(\IR^2\text{,}\) but the right example's image is a planar subspace of \(\IR^3\text{.}\)

Figure 3.3.8. The image of a linear transformation

Activity 3.3.9.

Let \(T: \IR^2 \rightarrow \IR^3\) be given by

\begin{equation*} T\left(\left[\begin{array}{c}x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x \\ y \\ 0 \end{array}\right] \hspace{3em} \text{with standard matrix } \left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right] \end{equation*}

Which of these subspaces of \(\IR^3\) describes \(\Im T\text{,}\) the set of all vectors that are the result of using \(T\) to transform \(\IR^2\) vectors?

  1. \(\displaystyle \setBuilder{\left[\begin{array}{c}0 \\ 0\\ a\end{array}\right]}{a\in\IR}\)

  2. \(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ b\\ 0\end{array}\right]}{a,b\in\IR}\)

  3. \(\displaystyle \setList{\left[\begin{array}{c}0\\0\\0\end{array}\right]}\)

  4. \(\displaystyle \IR^3=\setBuilder{\left[\begin{array}{c}x \\ y\\z\end{array}\right]}{x,y,z\in\IR}\)

Activity 3.3.10.

Let \(T: \IR^3 \rightarrow \IR^2\) be given by

\begin{equation*} T\left(\left[\begin{array}{c}x \\ y\\z \end{array}\right] \right) = \left[\begin{array}{c} x \\ y \end{array}\right] \hspace{3em} \text{with standard matrix } \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] \end{equation*}

Which of these subspaces of \(\IR^2\) describes \(\Im T\text{,}\) the set of all vectors that are the result of using \(T\) to transform \(\IR^3\) vectors?

  1. \(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ a\end{array}\right]}{a\in\IR}\)

  2. \(\displaystyle \setList{\left[\begin{array}{c}0\\0\end{array}\right]}\)

  3. \(\displaystyle \IR^2=\setBuilder{\left[\begin{array}{c}x \\ y\end{array}\right]}{x,y\in\IR}\)

Activity 3.3.11.

Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by the standard matrix

\begin{equation*} A = \left[\begin{array}{cccc} 3 & 4 & 7 & 1\\ -1 & 1 & 0 & 2 \\ 2 & 1 & 3 & -1 \end{array}\right] = \left[\begin{array}{cccc}T(\vec e_1)&T(\vec e_2)&T(\vec e_3)&T(\vec e_4)\end{array}\right] . \end{equation*}

Since \(T(\vec v)=T(x_1\vec e_1+x_2\vec e_2+x_3\vec e_3+x_4\vec e_4)\text{,}\) the set of vectors

\begin{equation*} \setList{ \left[\begin{array}{c}3\\-1\\2\end{array}\right], \left[\begin{array}{c}4\\1\\1\end{array}\right], \left[\begin{array}{c}7\\0\\3\end{array}\right], \left[\begin{array}{c}1\\2\\-1\end{array}\right] } \end{equation*}
  1. spans \(\Im T\)

  2. is a linearly independent subset of \(\Im T\)

  3. is a basis for \(\Im T\)

Observation 3.3.12.

Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by the standard matrix

\begin{equation*} A = \left[\begin{array}{cccc} 3 & 4 & 7 & 1\\ -1 & 1 & 0 & 2 \\ 2 & 1 & 3 & -1 \end{array}\right] . \end{equation*}

Since the set \(\setList{ \left[\begin{array}{c}3\\-1\\2\end{array}\right], \left[\begin{array}{c}4\\1\\1\end{array}\right], \left[\begin{array}{c}7\\0\\3\end{array}\right], \left[\begin{array}{c}1\\2\\-1\end{array}\right] }\) spans \(\Im T\text{,}\) we can obtain a basis for \(\Im T\) by finding \(\RREF A = \left[\begin{array}{cccc} 1 & 0 & 1 & -1\\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\) and only using the vectors corresponding to pivot columns:

\begin{equation*} \setList{ \left[\begin{array}{c}3\\-1\\2\end{array}\right], \left[\begin{array}{c}4\\1\\1\end{array}\right] } \end{equation*}

Activity 3.3.14.

Let \(T: \IR^3 \rightarrow \IR^4\) be the linear transformation given by the standard matrix

\begin{equation*} A = \left[\begin{array}{ccc} 1 & -3 & 2\\ 2 & -6 & 0 \\ 0 & 0 & 1 \\ -1 & 3 & 1 \end{array}\right] . \end{equation*}

Find a basis for the kernel and a basis for the image of \(T\text{.}\)

Activity 3.3.15.

Let \(T: \IR^n \rightarrow \IR^m\) be a linear transformation with standard matrix \(A\text{.}\) Which of the following is equal to the dimension of the kernel of \(T\text{?}\)

  1. The number of pivot columns

  2. The number of non-pivot columns

  3. The number of pivot rows

  4. The number of non-pivot rows

Activity 3.3.16.

Let \(T: \IR^n \rightarrow \IR^m\) be a linear transformation with standard matrix \(A\text{.}\) Which of the following is equal to the dimension of the image of \(T\text{?}\)

  1. The number of pivot columns

  2. The number of non-pivot columns

  3. The number of pivot rows

  4. The number of non-pivot rows

Observation 3.3.17.

Combining these with the observation that the number of columns is the dimension of the domain of \(T\text{,}\) we have the rank-nullity theorem:

The dimension of the domain of \(T\) equals \(\dim(\ker T)+\dim(\Im T)\text{.}\)

The dimension of the image is called the rank of \(T\) (or \(A\)) and the dimension of the kernel is called the nullity.

Activity 3.3.18.

Let \(T: \IR^3 \rightarrow \IR^4\) be the linear transformation given by the standard matrix

\begin{equation*} A = \left[\begin{array}{ccc} 1 & -3 & 2\\ 2 & -6 & 0 \\ 0 & 0 & 1 \\ -1 & 3 & 1 \end{array}\right] . \end{equation*}

Verify that the rank-nullity theorem holds for \(T\text{.}\)

Subsection 3.3.1 Videos

Figure 3.3.19. Video: The kernel and image of a linear transformation
Figure 3.3.20. Video: Finding a basis of the image of a linear transformation
Figure 3.3.21. Video: Finding a basis of the kernel of a linear transformation
Figure 3.3.22. Video: The rank-nullity theorem

Exercises 3.3.2 Exercises

Exercises available at checkit.clontz.org 1 .

https://checkit.clontz.org/#/banks/tbil-la/A3/