For the following matrices, write the kernel as the span of...

For the following matrices, write the kernel as the span of a finite number of vectors. Is the kernel a point, line, plane, or all of $\mathbb{R}^3$ ? (a) $\left(\begin{array}{ll}2 & -1\end{array}\right.$ 5), (b) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 3 & -2 & 0\end{array}\right)$, (c) $\left(\begin{array}{rrr}2 & 6 & -4 \\ -1 & -3 & 2\end{array}\right)$ (d) $\left(\begin{array}{rrr}1 & 2 & 5 \\ 0 & 4 & 8 \\ 1 & -6 & -11\end{array}\right)$ (e) $\left(\begin{array}{rrr}2 & -1 & 1 \\ -1 & 1 & -2 \\ 3 & -1 & 1\end{array}\right)$, (f) $\left(\begin{array}{rrr}1 & -2 & 3 \\ -3 & 6 & -9 \\ -2 & 4 & -6 \\ 3 & 0 & -1\end{array}\right)$

For the following matrices, write the kernel as the span of a finite number of vectors. Is the kernel a point, line, plane, or all of $\mathbb{R}^3$ ?
(a) $\left(\begin{array}{ll}2 & -1\end{array}\right.$ 5),
(b) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 3 & -2 & 0\end{array}\right)$,
(c)
$\left(\begin{array}{rrr}2 & 6 & -4 \\ -1 & -3 & 2\end{array}\right)$
(d)
$\left(\begin{array}{rrr}1 & 2 & 5 \\ 0 & 4 & 8 \\ 1 & -6 & -11\end{array}\right)$
(e) $\left(\begin{array}{rrr}2 & -1 & 1 \\ -1 & 1 & -2 \\ 3 & -1 & 1\end{array}\right)$,
(f)
$\left(\begin{array}{rrr}1 & -2 & 3 \\ -3 & 6 & -9 \\ -2 & 4 & -6 \\ 3 & 0 & -1\end{array}\right)$

Key Concepts

Dimension

The dimension of a subspace, such as the kernel, is the number of vectors in its basis. This number indicates the degrees of freedom within the kernel. A 0-dimensional kernel is just a point, a 1-dimensional kernel forms a line, a 2-dimensional kernel forms a plane, and if the kernel equals all of the space, its dimension equals the dimension of the space itself.

Geometric Interpretation

Understanding the kernel in geometric terms helps classify the type of solution set. For instance, if the kernel consists only of the zero vector, then it is a single point. If it forms a one-dimensional subspace, it is a line, a two-dimensional subspace is a plane, and in the case of all ?³, it is the entire space. This interpretation gives insight into the structure and properties of the linear transformation.

Kernel

The kernel, often called the null space, of a linear transformation or matrix is the set of all input vectors that the transformation sends to the zero vector. It is a subspace of the domain and represents the solutions to the homogeneous equation Ax = 0.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. When we describe a kernel as the span of a finite number of vectors, we are identifying a basis for that subspace, meaning any vector in the kernel can be expressed as a linear combination of the basis vectors.

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