pyroknife
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Suppose that A is an n × n matrix and u and v
are vectors in R^n. Show that if Au = Av and
u ≠ v, then A is not invertible.
This is the book's proof:
From the fact that Au = Av, we have A(u − v) = 0. If
A is invertible, then u − v = 0, that is, u = v, which
contradicts the statement that u = v.
Mine proof is, if A is invertible thus A^-1 exists
(A^-1)(Au)=(A^-1)(Av)
A^-1*A=I
→u=v
if u≠v then then the above does not hold, implying that A^-1 does not exist.
are vectors in R^n. Show that if Au = Av and
u ≠ v, then A is not invertible.
This is the book's proof:
From the fact that Au = Av, we have A(u − v) = 0. If
A is invertible, then u − v = 0, that is, u = v, which
contradicts the statement that u = v.
Mine proof is, if A is invertible thus A^-1 exists
(A^-1)(Au)=(A^-1)(Av)
A^-1*A=I
→u=v
if u≠v then then the above does not hold, implying that A^-1 does not exist.
