Find a vector x orthogonal to the row space of A, and a...

Find a vector $x$ orthogonal to the row space of $A$, and a vector $y$ orthogonat to the column space, and a vector $z$ orthogonal to the nullspace:
$$
A=\left[\begin{array}{lll}
1 & 2 & 1 \\
2 & 4 & 3 \\
3 & 6 & 4
\end{array}\right]
$$

Key Concepts

Column Space

The column space of a matrix is the span of its column vectors. It consists of all linear combinations of the columns and is central to concepts like the range of a linear transformation. The dimension of the column space is known as the column rank, and it provides insight into the solvability and consistency of linear systems.

Nullspace

The nullspace (or kernel) of a matrix is the set of all vectors that are mapped to the zero vector when multiplied by the matrix. This subspace captures the idea of solutions to the homogeneous equation Ax = 0, and its dimension, the nullity, is a key concept in understanding the degrees of freedom within the system.

Orthogonal Complement

The orthogonal complement of a subspace consists of all vectors that are orthogonal to every vector in that subspace. In the context of the fundamental subspaces of a matrix, the orthogonal complement relationships (such as the row space being the orthogonal complement of the nullspace in R^n) play a pivotal role in both theory and applications.

Left Nullspace

The left nullspace of a matrix is the set of all vectors that are orthogonal to its columns; it can also be described as the nullspace of the transpose of the matrix. This subspace is important when considering vectors orthogonal to the column space, and it helps in understanding the underlying structure of linear transformations.

Row Space

The row space of a matrix is the subspace spanned by its row vectors. In linear algebra, it represents all possible linear combinations of the rows, and its dimension is called the row rank of the matrix. Understanding the row space is crucial because it relates to the solutions of linear systems via the fundamental subspaces of a matrix.

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