Find bases for the four fundamental subspaces associated...

Find bases for the four fundamental subspaces associated with the complex matrices (a) $\left(\begin{array}{cc}\mathrm{i} & 2 \\ -1 & 2 \mathrm{i}\end{array}\right)$, (b) $\left(\begin{array}{ccc}2 & -1+\mathrm{i} & 1-2 \mathrm{i} \\ -4 & 3-\mathrm{i} & 1+\mathrm{i}\end{array}\right)$, (c) $\left(\begin{array}{ccc}\mathrm{i} & -1 & 2-\mathrm{i} \\ -1+2 \mathrm{i} & -2-\mathrm{i} & 3 \\ \mathrm{i} & -1 & 1+\mathrm{i}\end{array}\right)$.

Find bases for the four fundamental subspaces associated with the complex matrices
(a) $\left(\begin{array}{cc}\mathrm{i} & 2 \\ -1 & 2 \mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{ccc}2 & -1+\mathrm{i} & 1-2 \mathrm{i} \\ -4 & 3-\mathrm{i} & 1+\mathrm{i}\end{array}\right)$,
(c) $\left(\begin{array}{ccc}\mathrm{i} & -1 & 2-\mathrm{i} \\ -1+2 \mathrm{i} & -2-\mathrm{i} & 3 \\ \mathrm{i} & -1 & 1+\mathrm{i}\end{array}\right)$.

Key Concepts

Basis

A basis for a vector space is a set of vectors that are both linearly independent and span the entire space. In the context of the four fundamental subspaces, finding a basis means identifying the minimal set of vectors that can represent every element of each subspace, providing a compact and efficient description of the structure of the linear transformation.

Left Null Space

The left null space, sometimes called the cokernel, is the set of all vectors that annihilate the rows of the matrix (or, equivalently, all vectors y for which y^*A = 0 in the complex case). It is a subspace of the dual space to the codomain and can be determined by considering the null space of the conjugate transpose of the matrix. This subspace is fundamental in understanding the restrictions imposed by the matrix on the dual space.

Null Space

The null space of a matrix consists of all vectors that, when multiplied by the matrix, yield the zero vector. It is found by solving the homogeneous equation Ax = 0, and its basis is formed by the set of linearly independent solutions. The dimension of the null space, known as the nullity, provides insight into the degrees of freedom in the solution of linear systems.

Column Space

The column space of a matrix is the set of all possible linear combinations of its columns. It represents the image of the corresponding linear transformation and is a subspace of the codomain. A basis for the column space can typically be found by performing Gaussian elimination and selecting the pivot columns, and its dimension is equal to the rank of the matrix.

Fundamental Subspaces

The four fundamental subspaces associated with any matrix are the column space, row space, null space, and left null space. These subspaces provide a complete picture of the matrix as a linear transformation, detailing where the outputs lie (column space), the constraints on the inputs (null space), the behavior of the dual transformation (left null space), and how the rows combine to determine the image (row space).

Row Space

The row space is the set of all linear combinations of the matrix's rows and is a subspace of the domain. The basis for the row space is often obtained through Gaussian elimination or row-reduction, where the nonzero rows in reduced row echelon form serve as basis vectors. The row space is closely related to the column space of the transpose of the matrix.

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