Find a basis for the orthogonal complement of the row space...

Find a basis for the orthogonal complement of the row space of $A$ :
$$
A=\left[\begin{array}{lll}
1 & 0 & 2 \\
1 & 1 & 4
\end{array}\right]
$$
Split $x=(3,3,3)$ into a row space component $x_{r}$ and a nullspace component $x_{n}$.

Key Concepts

Basis

A basis is a minimal set of linearly independent vectors that span a vector space or subspace. Establishing a basis allows one to uniquely represent every element of the space as a linear combination of the basis vectors, simplifying many operations and analyses in linear algebra.

Row Space

The row space of a matrix is the set of all possible linear combinations of its row vectors. It is a subspace of the Euclidean space that the rows inhabit and plays a central role in understanding the solutions to linear equations, as well as determining the rank of the matrix.

Orthogonal Complement

The orthogonal complement of a subspace consists of all vectors that are perpendicular to every vector in the subspace. In the context of matrices, the orthogonal complement of the row space is the set of all vectors that yield a zero dot product with every row of the matrix.

Null Space

The null space of a matrix is the collection of all vectors that, when the matrix is multiplied by them, produce the zero vector. In many situations, particularly in finite-dimensional spaces, the null space is exactly the orthogonal complement of the row space.

Vector Decomposition

Vector decomposition involves splitting a given vector into two components, each belonging to orthogonal subspaces (for instance, one component in the row space and another in its orthogonal complement or null space). This process is essential for understanding projections and for solving problems in least squares and other applications in linear algebra.

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