sect7_4thms - A = rank A T 4. For n-component vectors, the...

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MATH 405: Matrices The Six Theorems of Sect.7.4 September 25, 2006 AB 1. Row reducing a matrix does not change its rank (does not really change the matrix) 2. To decide if p given vectors (each with n components) are linearly independent, use them as row vectors for a matrix A , and find its rank. Iff rank A = p then they are linearly independent (otherwise rank A < p ...) 3. Same thing for column vectors - so for any matrix rank
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Unformatted text preview: A = rank A T 4. For n-component vectors, the greatest number of linearly independent vectors is n 5. The vector space V of n-component vectors has dimension n (dim V = n ), which means it has n basis vectors. 6. The row space and the column space of a matrix A have the same dimension, which is equal to rank A ....
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