For each of the following matrices A,(i) find a basis for...

For each of the following matrices $A,(i)$ find a basis for each of the four fundamental subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify that the coimage and kernel are orthogonal complements: (a) $\left(\begin{array}{ll}1 & -2 \\ 2 & -4\end{array}\right)$, (b) $\left(\begin{array}{ll}5 & 0 \\ 1 & 2 \\ 0 & 2\end{array}\right)$, (c) $\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right)$, (d) $\left(\begin{array}{rrrr}1 & 2 & 0 & 1 \\ -1 & 1 & 3 & 1 \\ 0 & 3 & 3 & 2\end{array}\right)$, (e) $\left(\begin{array}{rrrrr}3 & 1 & 4 & 2 & 7 \\ 1 & 1 & 2 & 0 & 3 \\ 5 & 2 & 7 & 3 & 12\end{array}\right)$, (f) $\left(\begin{array}{rrrr}1 & 3 & 0 & -2 \\ -2 & 1 & 2 & 3 \\ -3 & 5 & 4 & 4 \\ 1 & -4 & -2 & -1\end{array}\right)$, (g) $\left(\begin{array}{rrrr}-1 & 2 & 2 & -1 \\ 2 & -4 & -5 & 2 \\ -3 & 6 & 2 & -3 \\ 1 & -2 & -3 & 1 \\ -2 & 4 & -5 & -2\end{array}\right)$.

For each of the following matrices $A,(i)$ find a basis for each of the four fundamental subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify that the coimage and kernel are orthogonal complements:
(a) $\left(\begin{array}{ll}1 & -2 \\ 2 & -4\end{array}\right)$,
(b) $\left(\begin{array}{ll}5 & 0 \\ 1 & 2 \\ 0 & 2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrrr}1 & 2 & 0 & 1 \\ -1 & 1 & 3 & 1 \\ 0 & 3 & 3 & 2\end{array}\right)$,
(e) $\left(\begin{array}{rrrrr}3 & 1 & 4 & 2 & 7 \\ 1 & 1 & 2 & 0 & 3 \\ 5 & 2 & 7 & 3 & 12\end{array}\right)$,
(f) $\left(\begin{array}{rrrr}1 & 3 & 0 & -2 \\ -2 & 1 & 2 & 3 \\ -3 & 5 & 4 & 4 \\ 1 & -4 & -2 & -1\end{array}\right)$,
(g) $\left(\begin{array}{rrrr}-1 & 2 & 2 & -1 \\ 2 & -4 & -5 & 2 \\ -3 & 6 & 2 & -3 \\ 1 & -2 & -3 & 1 \\ -2 & 4 & -5 & -2\end{array}\right)$.

Key Concepts

Four Fundamental Subspaces

This concept in linear algebra refers to the four key subspaces associated with any m×n matrix: the column space (range or image), the null space (kernel), the row space (coimage), and the left null space (cokernel). These subspaces are interrelated and provide a complete picture of the linear transformation defined by the matrix. Their dimensions are connected by the rank-nullity theorem, and they help in understanding the structure of the matrix and solving systems of linear equations.

Column Space

The column space of a matrix is the set of all linear combinations of its column vectors. It represents the image of the associated linear transformation, i.e., what outputs can be achieved. Finding a basis for the column space involves identifying a minimal set of independent columns that spans this space. This subspace is crucial when determining the rank and when discussing the solvability of Ax = b.

Null Space

The null space of a matrix is the set of all vectors that are mapped to the zero vector by the matrix. It characterizes the solutions of the homogeneous equation Ax = 0. A basis of the null space provides the structure of all possible solutions to this equation, and its dimension is given by the nullity of the matrix. Understanding the null space is essential for grasping the concept of redundancy in the system described by the matrix.

Row Space

The row space of a matrix is the set of all linear combinations of its row vectors, and it can be thought of as the image of the transpose of the matrix. A basis for the row space can be obtained by performing row operations, typically converting the matrix into a row echelon or reduced row echelon form. The row space is closely related to the column space and is used in analyzing the consistency of the linear system and the rank of the matrix.

Left Null Space

The left null space (or cokernel) of a matrix consists of all the vectors that, when multiplied by the matrix from the left, yield the zero vector. Conceptually, it is the null space of the transpose of the matrix. A basis for the left null space is important when assessing the constraints on the output of the matrix and plays a crucial role in orthogonal decomposition, particularly as the complement of the column space.

Orthogonal Complements

Orthogonal complements are pairs of subspaces where each vector in one subspace is orthogonal to every vector in the other, under a given inner product. In the context of the fundamental subspaces, the column space and the left null space are orthogonal complements, as are the row space and the null space. This relationship is a cornerstone in the understanding of the geometric structure of solutions to linear systems.

Basis

A basis for a vector space (or subspace) is a set of linearly independent vectors that span the space. In solving problems involving the four fundamental subspaces, one finds a basis for each to understand the dimensions and structure of these spaces. Choosing an appropriate basis simplifies computations and provides clear insight into the underlying linear transformation represented by the matrix.

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