Why are invertible matrices "one-to-one"?

1 Answer
Dec 20, 2015

Answer:

See explanation...

Explanation:

I think the question is referring to the natural use of a matrix to map points to points by multiplication.

Suppose #M# is an invertible matrix with inverse #M^(-1)#

Suppose further that #Mp_1 = Mp_2# for some points #p_1# and #p_2#.

Then multiplying both sides by #M^(-1)# we find:

#p_1 = I p_1 = M^(-1)M p_1 = M^(-1)M p_2 = I p_2 = p_2#

So:

#Mp_1 = Mp_2 => p_1 = p_2#

That is: multiplication by #M# is one-to-one.