Question-4

linear independence

basis

\(S\) is a linearly independent subset of \(\mathbb{R}^{4}\). Select all true statements concerning \(S\).

\(|S| \leqslant 4\)
Any subset of \(S\) is also a linearly independent subset of \(\mathbb{R}^{4}\).
Any superset of \(S\) is also a linearly independent subset of \(\mathbb{R}^{4}\).
\(S\) is a basis for \(\mathbb{R}^{4}\).

Answer

\(|S| \leqslant 4\)
Any subset of \(S\) is also a linearly independent subset of \(\mathbb{R}^{4}\).
Any superset of \(S\) is also a linearly independent subset of \(\mathbb{R}^{4}\).
\(S\) is a basis for \(\mathbb{R}^{4}\).

Solution

\(S\) will be a basis for \(\mathbb{R}^{4}\) if and only if \(|S| = 4\).