Let \[ \alpha_1 = (1, 1, -2, 1), \quad \alpha_2 = (3, 0, 4, -1), \quad \alpha_3 = (-1, 2, 5, 2). \] Let \[ \alpha = (4, -5, 9, -7), \quad \beta = (3, 1, -4, 4), \quad \gamma = (-1, 1, 0, 1). \]
Which of the vectors \(\alpha\), \(\beta\), \(\gamma\) are in the subspace of \(\RR^4\) spanned by the \(\alpha_i\)?
Which of the vectors \(\alpha\), \(\beta\), \(\gamma\) are in the subspace of \(C^4\) spanned by the \(\alpha_i\)?
Does this suggest a theorem?
Solution
1. The \(\alpha_i\) are linearly independent, so they span a \(3\)-dimensional subspace \(W\).
the matrix with columns \(\alpha_1, \alpha_2, \alpha_3\) has rank \(3\) (its three columns are independent).
2. A vector \(v\) lies in \(W\) iff the matrix \([\alpha_1\ \alpha_2\ \alpha_3\ v]\) has rank \(3\) (i.e. \(v\) does not increase the rank).
\(v \in W\) iff \(v\) is a linear combination of the \(\alpha_i\), iff the augmented matrix has the same rank as \([\alpha_1\ \alpha_2\ \alpha_3]\).
3. \(\alpha = (4,-5,9,-7)\) is in \(W\).
\(\operatorname{rank}[\alpha_1\ \alpha_2\ \alpha_3\ \alpha] = 3 = \operatorname{rank}[\alpha_1\ \alpha_2\ \alpha_3]\); indeed \(\alpha = -3\alpha_1 + 2\alpha_2 - \alpha_3\).
4. \(\beta = (3,1,-4,4)\) is not in \(W\).
\(\operatorname{rank}[\alpha_1\ \alpha_2\ \alpha_3\ \beta] = 4 > 3\).
5. \(\gamma = (-1,1,0,1)\) is not in \(W\).
\(\operatorname{rank}[\alpha_1\ \alpha_2\ \alpha_3\ \gamma] = 4 > 3\).
6. Since the \(\alpha_i\) have real entries, the real span and the complex span coincide on real vectors: a real vector is in the complex span iff it is in the real span.
if \(v = \sum c_i \alpha_i\) with \(c_i \in \CC\) and \(v, \alpha_i\) all real, then taking real parts gives \(v = \sum \operatorname{Re}(c_i) \alpha_i\) with real coefficients.
7. Hence the answers to (a) and (b) are identical: \(\alpha\) is in the span, \(\beta\) and \(\gamma\) are not.
<1>3–<1>6.
8. Theorem suggested: if \(v_1, \ldots, v_k \in \RR^n\) and \(v \in \RR^n\), then \(v\) is in the \(\CC\)-span of the \(v_i\) iff \(v\) is in the \(\RR\)-span of the \(v_i\).
6.
9. Q.E.D.
7 and 8.