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complements. Thus, if A is an matrix, to say that has only the trivial solution is equivalent to saying that the
orthogonal complement of the nullspace of A is all of , or, equivalently, that the rowspace of A is all of . This enables us
to add two new results to the seventeen listed in Theorem 5.6.9.
THEOREM 6.2.7
Equivalent Statements is multiplication by A, then the following are equivalent.
If A is an matrix, and if
(a) A is invertible.
(b) has only the trivial solution.
(c) The reduced row-echelon form of A is .
(d) A is expressible as a product of elementary matrices.
(e) is consistent for every matrix b.
(f) has exactly one solution for every matrix b.
(g) .
(h) The range of is .
(i) is one-to-one.
(j) The column vectors of A are linearly independent.
(k) The row vectors of A are linearly independent.
(l) The column vectors of A span .
(m) The row vectors of A span .
(n) The column vectors of A form a basis for .
