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
Activity 3.3.3.
Let \(T: \IR^2 \rightarrow \IR^3\) be given by
Which of these subspaces of \(\IR^2\) describes \(\ker T\text{,}\) the set of all vectors that transform into \(\vec 0\text{?}\)
\(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ a\end{array}\right]}{a\in\IR}\)
\(\displaystyle \setList{\left[\begin{array}{c}0\\0\end{array}\right]}\)
\(\displaystyle \IR^2=\setBuilder{\left[\begin{array}{c}x \\ y\end{array}\right]}{x,y\in\IR}\)
Activity 3.3.4.
Let \(T: \IR^3 \rightarrow \IR^2\) be given by
Which of these subspaces of \(\IR^3\) describes \(\ker T\text{,}\) the set of all vectors that transform into \(\vec 0\text{?}\)
\(\displaystyle \setBuilder{\left[\begin{array}{c}0 \\ 0\\ a\end{array}\right]}{a\in\IR}\)
\(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ a\\ 0\end{array}\right]}{a\in\IR}\)
\(\displaystyle \setList{\left[\begin{array}{c}0\\0\\0\end{array}\right]}\)
\(\displaystyle \IR^3=\setBuilder{\left[\begin{array}{c}x \\ y\\z\end{array}\right]}{x,y,z\in\IR}\)
Activity 3.3.5.
Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by the standard matrix
(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
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
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 \(\IR^3\text{.}\)
Activity 3.3.9.
Let \(T: \IR^2 \rightarrow \IR^3\) be given by
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?
\(\displaystyle \setBuilder{\left[\begin{array}{c}0 \\ 0\\ a\end{array}\right]}{a\in\IR}\)
\(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ b\\ 0\end{array}\right]}{a,b\in\IR}\)
\(\displaystyle \setList{\left[\begin{array}{c}0\\0\\0\end{array}\right]}\)
\(\displaystyle \IR^3=\setBuilder{\left[\begin{array}{c}x \\ y\\z\end{array}\right]}{x,y,z\in\IR}\)
Activity 3.3.10.
Let \(T: \IR^3 \rightarrow \IR^2\) be given by
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?
\(\displaystyle \setBuilder{\left[\begin{array}{c}a \\ a\end{array}\right]}{a\in\IR}\)
\(\displaystyle \setList{\left[\begin{array}{c}0\\0\end{array}\right]}\)
\(\displaystyle \IR^2=\setBuilder{\left[\begin{array}{c}x \\ y\end{array}\right]}{x,y\in\IR}\)
Activity 3.3.11.
Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by the standard matrix
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
spans \(\Im T\)
is a linearly independent subset of \(\Im T\)
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
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:
Fact 3.3.13.
Let \(T:\IR^n\to\IR^m\) be a linear transformation with standard matrix \(A\text{.}\)
The kernel of \(T\) is the solution set of the homogeneous system given by the augmented matrix \(\left[\begin{array}{c|c}A&\vec 0\end{array}\right]\text{.}\) Use the coefficients of its free variables to get a basis for the kernel.
The image of \(T\) is the span of the columns of \(A\text{.}\) Remove the vectors creating non-pivot columns in \(\RREF A\) to get a basis for the image.
Activity 3.3.14.
Let \(T: \IR^3 \rightarrow \IR^4\) be the linear transformation given by the standard matrix
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 \(T\text{?}\)
The number of pivot columns
The number of non-pivot columns
The number of pivot rows
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 \(T\text{?}\)
The number of pivot columns
The number of non-pivot columns
The number of pivot rows
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
Verify that the rank-nullity theorem holds for \(T\text{.}\)
Subsection 3.3.1 Videos
Exercises 3.3.2 Exercises
Exercises available at checkit.clontz.org 1 .
https://checkit.clontz.org/#/banks/tbil-la/A3/