## Section3.3Image and Kernel (AT3)

### Subsection3.3.1Class Activities

#### Activity3.3.1.

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

#### Definition3.3.2.

Let $$T: V \rightarrow W$$ be a linear transformation, and let $$\vec{z}$$ be the additive identity (the “zero vector”) of $$W\text{.}$$ 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*}

#### Activity3.3.3.

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

#### Activity3.3.4.

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 $$T\text{.}$$

#### Activity3.3.5.

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 $$T\text{.}$$

#### Activity3.3.6.

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

#### Definition3.3.7.

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 $$\IR^3\text{.}$$

#### Activity3.3.8.

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

#### Activity3.3.9.

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 for a vector $$\vec v =\left[\begin{array}{c}x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right] \text{,}$$ $$T(\vec v)=T(x_1\vec e_1+x_2\vec e_2+x_3\vec e_3+x_4\vec e_4)\text{,}$$ which of the following best describes 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] }\text{?} \end{equation*}
1. The set of vectors spans $$\Im T$$ but is not linearly independent.

2. The set of vectors is a linearly independent subset of $$\Im T$$ but does not span $$\Im T\text{.}$$

3. The set of vectors is linearly independent and spans $$\Im T\text{;}$$ that is, the set of vectors is a basis for $$\Im T\text{.}$$

#### Observation3.3.10.

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*}

#### Activity3.3.12.

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 $$T\text{.}$$

#### Activity3.3.13.

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

#### Activity3.3.14.

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

#### Observation3.3.15.

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.

#### Activity3.3.16.

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 $$T\text{.}$$

### Subsection3.3.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/AT3.slides.html.

### Subsection3.3.5Mathematical Writing Explorations

#### Exploration3.3.17.

Assume $$f:V \rightarrow W$$ is a linear map. Let $$\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}$$ be a set of vectors in $$V\text{,}$$ and set $$\vec{w_i} = f(\vec{v_i})\text{.}$$

• If the set $$\{\vec{w_1},\vec{w_2},\ldots,\vec{w_n}\}$$ is linearly independent, must the set $$\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}$$ also be linearly independent?

• If the set $$\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}$$ is linearly independent, must the set $$\{\vec{w_1},\vec{w_2},\ldots,\vec{w_n}\}$$ also be linearly independent?

• If the set $$\{\vec{w_1},\vec{w_2},\ldots,\vec{w_n}\}$$ spans $$W\text{,}$$ must the set $$\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}$$ also span $$V\text{?}$$

• If the set $$\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}$$ spans $$V\text{,}$$ must the set $$\{\vec{w_1},\vec{w_2},\ldots,\vec{w_n}\}$$ also span $$W\text{?}$$

• In light of this, is the image of the basis of a vector space always a basis for the codomain?

#### Exploration3.3.18.

• The theorem states that, given a linear map $$h:V \rightarrow W\text{,}$$ with $$V$$ and $$W$$ vector spaces, the rank of $$h\text{,}$$ plus the nullity of $$h\text{,}$$ equals the dimension of the domain $$V\text{.}$$ Assume that the dimension of $$V$$ is $$n\text{.}$$

• For simplicity, denote the rank of $$h$$ by $$\mathcal{R}(h)\text{,}$$ and the nullity by $$\mathcal{N}(h)\text{.}$$

• Recall that $$\mathcal{R}(h)$$ is the dimension of the range space of $$h\text{.}$$ State the precise definition.

• Recall that $$\mathcal{N}(h)$$ is the dimension of the null space of $$h\text{.}$$ State the precise definition.

• Begin with a basis for the null space, denoted $$B_N = \{\vec{\beta_1}, \vec{\beta_2}, \ldots, \vec{\beta_k}\}\text{.}$$ Show how this can be extended to a basis $$B_V$$ for $$V\text{,}$$ with $$B_V = \{\vec{\beta_1}, \vec{\beta_2}, \ldots, \vec{\beta_k}, \vec{\beta_{k+1}}, \vec{\beta_{k+2}}, \ldots, \vec{\beta_n}\}.$$ In this portion, you should assume $$k \leq n\text{,}$$ and construct additional vectors which are not linear combinations of vectors in $$B_N\text{.}$$ Prove that you can always do this until you have $$n$$ total linearly independent vectors.

• Show that $$B_R = \{h(\vec{\beta_{k+1}}), h(\vec{\beta_{k+2}}), \ldots, h(\vec{\beta_n})\}$$ is a basis for the range space. Start by showing that it is linearly independent, and be sure you prove that each element of the range space can be written as a linear combination of $$B_R\text{.}$$

• Show that $$B_R$$ spans the range space.