One-to-one linear transformations

[Nikitin]

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...

 

[micromass]

Hint: http://en.wikipedia.org/wiki/Rank–nullity_theorem

 

[Nikitin]

Is there no alternative to insanely difficult wikipedia proofs?

 

[micromass]

What does your textbook say? What is your textbook?

Do they prove the rank-nullity theorem??

 

[jbunniii]

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]

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.