I sort of recognize one to one, but I cannot, for the life of me, recognize anything about if a matrix is onto. Every definition I look at is really difficult to understand.

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The summary

*onto*(or

*surjective*) go not use to matrices only, therefore I'll simply lay out the full definition.

**Def.** A function *f* indigenous *A* come *B* is dubbed *onto* if for every *b*∈*B* over there is *a*∈*A* such that *f*(*a*) = *b*.

In other words, *f* is *onto* if there is no point in *B* that have the right to not be reached as *f*(*a*). For example, if we take into consideration functions **R**↦**R** (that map from the real line to the genuine line), we have:

*f*(*x*) = *x*2 **is not** onto, because no issue which actual number you put right into *f*, you can never acquire a an unfavorable real number out. In other words, there space some number in the codomain the cannot it is in reached. (Remember the for a duty *f*:*A*↦*B*, *A* is referred to as the domain and *B* the codomain.)

*f*(*x*) = *x*3 **is** onto, since for any number *x* (negative, optimistic or zero) the cube root is fine defined and also real, so you deserve to reach any real number through inputting one more real number right into *f*.

*f*(*x*) = 1 **is not** onto, due to the fact that no matter what you put in, you deserve to only with the number 1. No other output is possible.

*f*(*x*) = sin(*x*) **is not** onto, due to the fact that you can only acquire numbers in <-1, 1> together output. No other outputs room possible.

*f*(*x*) = *ax*+*b* (assuming *a* is nonzero) **is** onto, due to the fact that if you want to reach the calculation number *y*, simply let *x*=(*y*-*b*)/*a*.

So anyway, ago to matrices. The reason we have the right to use words onto because that matrices is the a procession *M* is a representation of a direct map ~ you settle a basis. A procession *M* is then taken into consideration onto if the straight map it represents is onto. If *linear map* and *basis* don't sound familiar, here's what this means for matrices. A *m*×*n* matrix *M* is favor a duty that maps vectors native **R***n* to **R***m* by way of multiplication. So if **v** is in **R***n* climate *M***v** is in **R***m*. The matrix *M* is then onto if every element in **R***m* deserve to be reached from some element in **R***n* by multiplying v *M*.

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A simple means to test if a *m*×*n* procession *M* is ~ above is to minimize it to heat echelon form and examine for a heat that consists of just zeroes. If over there is together a row, the matrix is **not** onto. However, if every rows room nonzero, the procession is onto. You deserve to think the this as having to carry out with levels of freedom. If only *m*-1 rows space nonzero (so there's one zero row), then you have actually only *m*-1 levels of flexibility in specifying what value *M***v** must take if you gain to select **v** freely. In various other words, the collection of reachable vectors is *m*-1 dimensional. In other words, it's not **R***m* (because **R***m* is *m* dimensional). The very same intuition holds if over there is more than one nonzero row.