How can I show that the rank of a matrix is not altered if a column is multiplied by a non-zero scalar? I don't know where to start.
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this is the effect of multipplying on the right by a square matrix, the identity except one diagonal element is replaced by your non-zero scalar – Will Jagy Nov 08 '20 at 17:09
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@Will Jagy I didn't get it. – Program Nov 08 '20 at 17:11
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rank = dim of column space – Lozenges Nov 08 '20 at 17:15
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The dimension of the space spanned by the row vectors is the same as that spanned by its column vectors. Let $C = \{c_1, \dots, c_n\}$ be the collection of $n$ column vectors. Its span is unchanged if we multiply one of the $c_i$ by a nonzero scalar. So whether you define rank using rows or columns, you have not changed the rank.
Eric Towers
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