1.I.1H

Linear Algebra | Part IB, 2004

Suppose that {e1,,er+1}\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r+1}\right\} is a linearly independent set of distinct elements of a vector space VV and {e1,,er,fr+1,,fm}\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{r}, \mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m}\right\} spans VV. Prove that fr+1,,fm\mathbf{f}_{r+1}, \ldots, \mathbf{f}_{m} may be reordered, as necessary, so that {e1,er+1,fr+2,,fm}\left\{\mathbf{e}_{1}, \ldots \mathbf{e}_{r+1}, \mathbf{f}_{r+2}, \ldots, \mathbf{f}_{m}\right\} spans VV.

Suppose that {e1,,en}\left\{\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}\right\} is a linearly independent set of distinct elements of VV and that {f1,,fm}\left\{\mathbf{f}_{1}, \ldots, \mathbf{f}_{m}\right\} spans VV. Show that nmn \leqslant m.

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