You are watching: How to tell if a matrix is onto

This is the function I have:

How deserve to I examine if it"s one come one? perform I require to examine for everything that is in my function T, therefore let"s speak 3x-z, and also take a random number for x and z, and also check whether the worth that rolls out, is *unique*? i did this, however I don"t think this is exactly to be honest.

What is the method to technique this? I"d really evaluate some assist and tips...

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edited Mar 17 "17 in ~ 11:29

Siyah

request Mar 16 "17 at 16:59

SiyahSiyah

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## 3 answers 3

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T is one-to-one duty if the kernel is trivial, so friend you neeed to check implication $T(x,y,z)=0implies x=y=z=0$.

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answer Mar 16 "17 at 17:04

alansalans

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Find the REF that the traditional matrix (it"s not necessary to acquire to RREF). Then, look in ~ the pivots (the top 1"s the the rows).

If we have actually a pivot in every column, then the nullspace of the procession (and hence the kernel the $T$) is zero-dimensional. So, $T$ is one-to-one if and also only if the REF has actually pivot in every column.

If we have a pivot in every row, then $T(x,y,z) = wcchristchurchappeal.combf b$ has a solution for any kind of $ wcchristchurchappeal.combf b$. The is, the measurement of the photo is same to the dimension of the space. So, $T$ is ~ above if and also only if the REF has a pivot in every row.

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reply Mar 16 "17 at 17:15

Ben GrossmannBen Grossmann

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alans has provided you the technique for b)

One that the countless ways to fix for c) :

Find the selection of the linear transformation. I.e. $R(T) = spanT(1,0,0),T(0,1,0),T(0,0,1) = span(3,1,1,0),(0,2,-1,1),(-1,0,1,2)$

Note: You have the right to use any other communication you like. Also, i would certainly strongly introduce you to prove the the over statement is true in general.

Suppose top top the contrary that $R(T)$ is equal to $ wcchristchurchappeal.combbR^4$.

Then over there exist $lambda_1,lambda_2,lambda_3in wcchristchurchappeal.combbR$ such the $(0,0,0,1) = lambda_1cdot (3,1,1,0) + lambda_2cdot (0,2,-1,1) + lambda_3cdot (-1,0,1,2)$

Now, this mechanism of straight equation gives you no solution, i m sorry is a contradiction.

Note: mean you are given one more example i beg your pardon $T$ is without doubt onto, climate you will likewise find $R(T)$ and also show that it is equal to the vector room that $T$ is mapped to.

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edited Mar 17 "17 in ~ 18:03

reply Mar 17 "17 in ~ 17:47

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