For each matrix A find vectors that span the kernel of A....

For each matrix A find vectors that span the kernel of A. Use paper and pencil.
$$A=\left[\begin{array}{rrrrr}
1 & -1 & -1 & 1 & 1 \\
-1 & 1 & 0 & -2 & 2 \\
1 & -1 & -2 & 0 & 3 \\
2 & -2 & -1 & 3 & 4
\end{array}\right]$$

Key Concepts

Span of Vectors

The span of a set of vectors is the collection of all possible linear combinations of those vectors. When finding a basis for the kernel of a matrix, the chosen spanning vectors must be capable of generating every element of the kernel, thereby characterizing the solution space completely.

Kernel of a Matrix

The kernel (or null space) of a matrix is the set of all vectors that satisfy the equation Ax = 0. It is a fundamental concept in linear algebra that describes the solutions to homogeneous systems of linear equations and highlights the dependency relations between the columns of a matrix.

Homogeneous Systems

A homogeneous system of linear equations, represented as Ax = 0, always has at least the trivial solution. The process of finding all solutions involves determining the kernel of the matrix, which yields a description of all vectors that nullify the transformation defined by A.

Gaussian Elimination

Gaussian elimination (or row reduction) is a systematic method for simplifying a matrix to row-echelon or reduced row-echelon form. This process is crucial for solving systems of equations, as it helps in identifying pivot and free variables, thereby facilitating the determination of the kernel.

Free and Pivot Variables

In the context of solving a system of linear equations, pivot variables are those associated with leading entries in the row-reduced matrix, while free variables can take arbitrary values. The relationships between these variables allow one to express the general solution to the system, typically leading to a parametric description of the kernel.

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