Find a basis of the solution space of the following...

Find a basis of the solution space of the following homogeneous linear systems. (a) $$ \begin{aligned} x_1-2 x_3 & =0, \\ x_2+x_4 & =0 . \end{aligned} $$ (b) $$ \begin{array}{r} 2 x_1+x_2-3 x_3+x_4=0, \\ 2 x_1-x_2-x_3-x_4=0 . \end{array} $$ $$ \begin{array}{r} x_1-x_2-2 x_3+4 x_4=0, \\ 2 x_1+x_2-x_4=0, \\ -2 x_1+2 x_3-2 x_4=0 . \end{array} $$

Find a basis of the solution space of the following homogeneous linear systems.
(a)
$$
\begin{aligned}
x_1-2 x_3 & =0, \\
x_2+x_4 & =0 .
\end{aligned}
$$
(b)
$$
\begin{array}{r}
2 x_1+x_2-3 x_3+x_4=0, \\
2 x_1-x_2-x_3-x_4=0 .
\end{array}
$$
$$
\begin{array}{r}
x_1-x_2-2 x_3+4 x_4=0, \\
2 x_1+x_2-x_4=0, \\
-2 x_1+2 x_3-2 x_4=0 .
\end{array}
$$

Key Concepts

Gaussian Elimination / Row Reduction

Gaussian elimination is the process of applying elementary row operations to simplify a system of linear equations to row echelon or reduced row echelon form. This process is used to identify pivot variables and free variables, which is a key step in determining a basis for the solution space.

Free Variables and Parameterization

When solving homogeneous linear systems, some variables can be designated as free variables, meaning they can take on arbitrary values. Expressing the other variables (dependent variables) in terms of these free variables leads to a parameterized description of the solution space, which naturally indicates a basis.

Null Space (Kernel) of a Matrix

The null space of a matrix consists of all the vectors that are mapped to the zero vector when multiplied by the matrix. In the context of a homogeneous linear system, the solution space is exactly the null space, and understanding its structure and dimension is fundamental in linear algebra.

Homogeneous Linear System

A homogeneous linear system is one in which all of the constant terms are zero. This means that the system always has at least the trivial solution (where all variables equal zero), and the set of all solutions forms a subspace of the corresponding vector space.

Basis of a Vector Space

A basis for a vector space is a set of vectors that are both linearly independent and span the entire space. Finding a basis for the solution space means identifying a minimal set of vectors such that every solution of the homogeneous system can be expressed as a linear combination of these vectors.

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