one-to-one linear transformations


by Nikitin
Tags: linear, onetoone, transformations
Nikitin
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#1
Feb24-13, 08:37 AM
P: 588
Why is a linear transformation T(x)=Ax one-to-one if and only if the columns of A are linearly independent?

I don't get it...
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micromass
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#2
Feb24-13, 09:14 AM
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Hint: http://en.wikipedia.org/wiki/Rank%E2...ullity_theorem
Nikitin
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#3
Feb24-13, 11:54 AM
P: 588
Is there no alternative to insanely difficult wikipedia proofs?

micromass
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#4
Feb24-13, 11:55 AM
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one-to-one linear transformations


What does your textbook say? What is your textbook?

Do they prove the rank-nullity theorem??
jbunniii
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#5
Feb24-13, 02:33 PM
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T is one-to-one if and only if T(x) = T(y) implies x = y, if and only if T(x-y) = 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.
Nikitin
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#6
Feb24-13, 05:05 PM
P: 588
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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