Linear Transformation is Injective iff Kernel Contains Only Zero – ProofWiki

New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of mathsf{Pr} infty mathsf{fWiki}. If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code. is injective if and only if operatorname {N} left({mathbf A}right) = left{ {mathbf 0}right} That mathbf 0 in ker left({T}right) follows from Kernel of Linear Transformation contains Zero Vector. From Matrix Product as Linear Transformation, mathbf x mapsto mathbf {Ax} defines a linear transformation. forall mathbf x_1,mathbf x_2 in mathbf V: T left({mathbf x_1}right) = T left({mathbf x_2}right) implies mathbf x_1 = mathbf x_2 Source.


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