one to one and onto transformations

One to one definition

T: R^n → Rm is one to one if T(x) = b has at most one solution for all b in R^m

How to check one to one

Write T(x) = Ax. If A has a pivot is every column then T is one to one

How to show NOT one to one

find two different vectors that T maps to the same vector

Onto definition

T: R^n to R^m is onto if T(x) = b has at least one solution for all b in R^m

How to check onto

write T(x) = A(x). If A has a pivot in every row, then T is onto

If the row echelon form of A has a pivot in every column, is the transformation onto?

There is not enough information to tell whether the transformation T is onto because we do not know if there is a pivot in every row

If two rows in the row echelon form of A do not have pivots, is the transformation onto?

No, because two rows of the RREF are 0, so Ax=b must fail to be consistent for some b.

T is an onto transformation from R³ to R³

For every y in R³, there is an x in R³ such that T(x) = y. In other words, everything in the codomain is the image of something in the domain under T

T is a transformation from R³ to R³

For every x in R³, there is a y in R³ such that T(x) = y. This just says that every input to T has an output

T is a one to one transformation from R³ to R³

for every y in R³, there is at most one x in R³ such that T(x) = y. This means two different x in R³ can’t map to the same y under T.