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Onetoone linear transformations 
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#1
Feb2413, 08:37 AM

P: 627

Why is a linear transformation T(x)=Ax onetoone if and only if the columns of A are linearly independent?
I don't get it... 


#2
Feb2413, 09:14 AM

Mentor
P: 18,333



#3
Feb2413, 11:54 AM

P: 627

Is there no alternative to insanely difficult wikipedia proofs?



#4
Feb2413, 11:55 AM

Mentor
P: 18,333

Onetoone linear transformations
What does your textbook say? What is your textbook?
Do they prove the ranknullity theorem?? 


#5
Feb2413, 02:33 PM

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P: 3,288

T is onetoone if and only if T(x) = T(y) implies x = y, if and only if T(xy) = 0 implies x  y = 0, if and only if T(v) = 0 implies v = 0. But T(v) is a linear combination of the columns of A, so this says the only way to combine the columns of A to get zero is if the vector of coefficients (v) is zero. In other words, the columns of A are linearly independent.



#6
Feb2413, 05:05 PM

P: 627

Ahh, thanks Jbunny. It makes perfect sense now!
Micromass: Yes, it was, but the proofs in my book are written similarly to wikipedia  very tiresomely. 


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