Math 333
Hopefully this exercise will help you understand how most of the things we have learned are
related. This expands on the Big Theorem we learned in class involving square matrices by
considering non-square matrices.
Exercise: Read through the following statements. Determine which statements are equiv-
alent, and group equivalent statements together. You should be able to explain why they
are equivalent, but do not prove the equivalence (unless you want to.) Every statement
is equivalent to at least one other statement. Let Abe an m×nmatrix and let Tbe the
linear transformation given by T(x) = Ax.
(a) Ax=bhas at least one solution for any b.
(b) The reduced row echelon form of Ais the identity matrix.
(c) Tis one-to-one, but not necessarily onto.
(d) Ahas a free column, and the rows of Aare linearly independent.
(e) All rows of Aare linearly independent, and all columns of Aare linearly independent.
(f) Ais invertible.
(g) Tis onto, but not necessarily one-to-one.
(h) Ax=bhas at most one solution for any b.
(i) Tis both one-to-one and onto.
(j) Ax=0has only the trivial solution.
(k) Ahas mpivot columns.
(l) Tis invertible.
(m) The null space of Ais trivial.
(n) Ax=bhas exactly one solution for any b.
(o) The columns of Aare linearly independent.
(p) Ax=0has only the trivial solution, and m=n.
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