Consider the vectors in \(\RR^4\) defined by \[ \alpha_1 = (-1, 0, 1, 2), \quad \alpha_2 = (3, 4, -2, 5), \quad \alpha_3 = (1, 4, 0, 9). \] Find a system of homogeneous linear equations for which the space of solutions is exactly the subspace of \(\RR^4\) spanned by the three given vectors.
Solution
Goal. Find homogeneous linear equations whose solution space is \(W = \operatorname{span}\theset{\alpha_1, \alpha_2, \alpha_3}\).
1. Compute the dimension of \(W\).
1.1. Form the matrix \(A\) with rows \(\alpha_1, \alpha_2, \alpha_3\) and row-reduce.
row-reduce to find the rank.
1.2. \(\operatorname{rank} A = 2\).
row-reducing \(\begin{pmatrix} -1 & 0 & 1 & 2 \\ 3 & 4 & -2 & 5 \\ 1 & 4 & 0 & 9 \end{pmatrix}\) gives rank \(2\) (the three vectors are linearly dependent, spanning a \(2\)-dimensional subspace).
1.3. Hence \(\dim W = 2\), so \(W\) is cut out by \(4 - 2 = 2\) independent linear equations.
the solution space of a system of rank \(r\) in \(\RR^4\) has dimension \(4 - r\), so we need \(r = 2\).
2. Find a basis for \(W^\perp\).
2.1. \(W^\perp\) is the null space of \(A\) (vectors \(n\) with \(n \cdot \alpha_i = 0\) for all \(i\)).
\(W^\perp = \theset{n : n \cdot \alpha_i = 0 \text{ for } i = 1,2,3}\).
2.2. A basis for \(W^\perp\) is \(\theset{n_1, n_2}\) with \(n_1 = (4, -1, 4, 0)\) and \(n_2 = (8, -11, 0, 4)\).
solving \(A n = 0\) gives the null space spanned by \((1, -1/4, 1, 0)\) and \((2, -11/4, 0, 1)\); clearing denominators gives \(n_1 = (4, -1, 4, 0)\) and \(n_2 = (8, -11, 0, 4)\).
3. The system is \(n_1 \cdot x = 0\) and \(n_2 \cdot x = 0\).
3.1. \(W = \theset{x \in \RR^4 : n_1 \cdot x = 0 \text{ and } n_2 \cdot x = 0}\).
\(W\) is \(2\)-dimensional, and the two equations \(n_1 \cdot x = 0\), \(n_2 \cdot x = 0\) cut out a \(2\)-dimensional subspace containing \(W\) (since \(n_1, n_2 \perp W\)), so they are equal.
3.2. Explicitly, the system is\[ \begin{cases} 4x_1 - x_2 + 4x_3 = 0, \\ 8x_1 - 11x_2 + 4x_4 = 0. \end{cases} \]
expand \(n_1 \cdot x = 0\) and \(n_2 \cdot x = 0\).
4. Q.E.D.
3.2 gives the homogeneous system whose solution space is exactly \(W\).