For each of the following matrices find bases for the (i)...

For each of the following matrices find bases for the (i) image, (ii) coimage, (iii) kernel, and (iv) cokernel. (a) $\left(\begin{array}{ll}1 & -3 \\ 2 & -6\end{array}\right)$, (b) $\left(\begin{array}{rrr}0 & 0 & -8 \\ 1 & 2 & -1 \\ 2 & 4 & 6\end{array}\right)$, (c) $\left(\begin{array}{rrrr}1 & 1 & 2 & 1 \\ 1 & 0 & -1 & 3 \\ 2 & 3 & 7 & 0\end{array}\right)$, (d) $\left(\begin{array}{rrrrr}1 & -3 & 2 & 2 & 1 \\ 0 & 3 & -6 & 0 & -2 \\ 2 & -3 & -2 & 4 & 0 \\ 3 & -3 & -6 & 6 & 3 \\ 1 & 0 & -4 & 2 & 3\end{array}\right)$.

For each of the following matrices find bases for the (i) image, (ii) coimage, (iii) kernel, and (iv) cokernel.
(a) $\left(\begin{array}{ll}1 & -3 \\ 2 & -6\end{array}\right)$,
(b) $\left(\begin{array}{rrr}0 & 0 & -8 \\ 1 & 2 & -1 \\ 2 & 4 & 6\end{array}\right)$,
(c) $\left(\begin{array}{rrrr}1 & 1 & 2 & 1 \\ 1 & 0 & -1 & 3 \\ 2 & 3 & 7 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrrrr}1 & -3 & 2 & 2 & 1 \\ 0 & 3 & -6 & 0 & -2 \\ 2 & -3 & -2 & 4 & 0 \\ 3 & -3 & -6 & 6 & 3 \\ 1 & 0 & -4 & 2 & 3\end{array}\right)$.

Key Concepts

Basis

A basis for a vector space or subspace is a set of linearly independent vectors that span the entire space. Choosing a basis for subspaces such as the image, kernel, coimage, and cokernel simplifies many problems in linear algebra, as it reduces the problem to considering independent directions and provides a coordinate system for further computations.

Cokernel

The cokernel of a linear transformation is the quotient of the codomain by the image. It measures the extent to which the image fails to cover the entire codomain. A basis for the cokernel provides a means of understanding the difference between the ambient codomain and the part that is actually reached by the transformation.

Kernel

The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain. For a matrix, it is known as the null space. Finding a basis for the kernel means identifying the set of independent solutions to the homogeneous equation Ax = 0. The kernel provides insight into the 'loss of information' or the redundancy in the mapping process.

Image

The image of a linear transformation (or matrix) is the set of all vectors that can be reached as outputs by applying the transformation to vectors in the domain. In the case of matrices, it is often identified with the column space, which is spanned by the columns of the matrix. Determining a basis for the image involves identifying a minimal set of linearly independent columns that span this output space.

Coimage

The coimage of a linear transformation refers to the factor space of the domain by the kernel, essentially describing the effective portion of the domain on which the transformation acts injectively. It represents the structure of the domain after all the vectors that vanish under the transformation have been collapsed to zero. In many contexts, it is isomorphic to the image, as described by the first isomorphism theorem.

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