In Exercises 5-12, a matrix A is given. Determine the...

In Exercises 5-12, a matrix A is given. Determine the dimension of (a) Col A, (b) Null A, (c) Row A, and (d) Null A ${ }^T$.
$\left[\begin{array}{rrr}1 & 1 & 1 \\ 1 & -1 & 5 \\ 2 & 1 & 4 \\ 0 & 2 & -4\end{array}\right]$

Key Concepts

Column Space

The column space of a matrix, sometimes called the range, is the set of all possible linear combinations of its columns. It reflects the set of outputs the matrix can produce and its dimension, known as the rank, indicates the number of linearly independent columns of the matrix.

Null Space

The null space of a matrix is the set of all vectors that are mapped to the zero vector by the matrix transformation. Its dimension, called the nullity, shows how many free parameters exist in the solutions to the homogeneous equation Ax = 0.

Row Space

The row space of a matrix is the set of all linear combinations of its rows. It is important because the row space of A is equal to the column space of A transposed, and its dimension is also equal to the rank of the matrix.

Left Null Space

The left null space, defined as the null space of the transpose of the matrix, consists of all vectors that yield the zero vector when pre-multiplied by the matrix. Its dimension is related to the number of rows of the matrix and complements the rank in the context of the matrix's full row space.

Rank-Nullity Theorem

The rank-nullity theorem is a fundamental result in linear algebra that relates the dimensions of the various subspaces of a matrix. It states that for a matrix, the sum of the rank (dimension of the column space) and the nullity (dimension of the null space) equals the number of columns, thereby linking the structure of the solution space of the matrix equation to its columnar properties.

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