Let A be a m × n matrix with entries in R. Recall that the row...

Let A be a m × n matrix with entries in R. Recall that the row rank of A means the dimension of the subspace in R^N spanned by the rows of A (viewed as vectors in Rⁿ), and the column rank means that of the subspace in R^m spanned by the columns of A (viewed as vectors in R^m).

(a) Prove that

n = (column rank of A) + dim S,

where the set S is the solution space of the homogeneous equation AX = 0, that is, S = {column vectors X : AX = 0} .

(b) Show that

row rank of A = column rank of A.

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Expert Solution

Colby Messinger answered 1 year ago

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