The rank of an n × n matrix A is the maximum number of linearly
independent columns in A. Column vectors of of A, denoted
ai, are linearly independent if the only set of scalars
such that
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(23) |
is the set
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For a more concrete example, consider the following matrix.
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The rank of A is two, since its third and fourth columns are linear combinations
of its first two columns, i.e.
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If A is an n × n matrix, it can be shown
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(24) |
Furthermore,
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(25) |
and
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(26) |