Without computing A, find bases for the four fundamental...

Without computing $A$, find bases for the four fundamental subspaces:
$$
A=\left[\begin{array}{lll}
1 & 0 & 0 \\
6 & 1 & 0 \\
9 & 8 & 1
\end{array}\right]\left[\begin{array}{llll}
1 & 2 & 3 & 4 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{array}\right]
$$

Key Concepts

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 linear transformation and defines the target subspace into which vectors are mapped. The basis for the column space gives insight into the span and the dimension (rank) of the matrix, which is directly related to the solvability of the associated system of linear equations.

Fundamental Subspaces

In linear algebra, the four fundamental subspaces associated with any matrix provide a complete picture of the linear transformation the matrix represents. They include the column space (range), nullspace (kernel), row space, and left nullspace. These subspaces help in understanding the structure of solutions to linear systems, the behavior of the matrix transformation, and important properties like rank and nullity. They are interconnected through the Fundamental Theorem of Linear Algebra, which highlights the orthogonal relationships among them.

Nullspace

The nullspace of a matrix is the set of all vectors that are mapped to the zero vector by the transformation, meaning these vectors satisfy the homogeneous equation. It reveals the amount of redundancy or dependency in the system, with its dimension (nullity) indicating the number of free parameters in the solution. This concept is vital in understanding the degree of freedom and the solution space for homogeneous linear systems.

Row Space

The row space is formed by the span of the rows of a matrix and, equivalently, is the column space of the transposed matrix. It reflects the linear independence of the equations represented by the matrix and is crucial for understanding the relationships between different equations in a system. Additionally, the row space shares the same dimension as the column space, and this fact is central to the Rank-Nullity Theorem.

Left Nullspace

The left nullspace, also known as the cokernel of the matrix, consists of all vectors that, when multiplied from the left, yield the zero vector. It serves as the orthogonal complement to the row space in the domain of the matrix and plays a significant role in characterizing the constraints imposed by the matrix on its domain. This subspace is important for identifying compatibility conditions in linear systems and for understanding the structure of solutions to the corresponding adjoint problems.

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