A First Course in Linear Algebra · Problem ILT-M60
Linear Transformations — Robert Beezer
Mathematics — Linear Algebra · injective-linear-transformations
Problem
Suppose and are vector spaces. Define the function by for every . Then by Exercise LT.M60, is a linear transformation. Formulate a condition on that is equivalent to being an injective linear transformation. In other words, fill in the blank to complete the following statement (and then give a proof): is injective if and only if is \_\_\_\_\_\_\_\_\_\_. (See Exercise SLT.M60, Exercise IVLT.M60.)Intuitive explanation
The zero map sends every vector to the zero vector, so it's injective only when there's nothing "extra" to map—when the domain itself is trivial. Think about injectivity: a function is injective when different inputs give different outputs. Here, all inputs give the same output (), so the only way to avoid collisions is if there's only one input to begin with. This means must contain only the zero vector, making it the trivial vector space.
Intuitive explanation
The zero map sends every vector to the zero vector, so it's injective only when there's nothing "extra" to map—when the domain itself is trivial. Think about injectivity: a function is injective when different inputs give different outputs. Here, all inputs give the same output (), so the only way to avoid collisions is if there's only one input to begin with. This means must contain only the zero vector, making it the trivial vector space.