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Introduction
In this chapter, we deepen our exploration of vector spaces through a widespread technique in mathematics: in order to understand the structure of mathematical objects, we study functions between those objects. In this case, we introduce
linear transformations as structure-preserving functions between spaces of vectors. We begin here by taking a mostly algebraic approach, deferring geometric explorations for later chapters.
Motivating Question. How can we understand linear maps algebraically?
Learning Outcomes
By the end of this chapter, you should be able to...
Determine if a map between Euclidean vector spaces is linear or not.
Translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.
Compute a basis for the kernel and a basis for the image of a linear map, and verify that the rank-nullity theorem holds for a given linear map.
Determine if a given linear map is injective and/or surjective.
Explain why a given set with defined addition and scalar multiplication does satisfy a given vector space property, but nonetheless isn’t a vector space.
Answer questions about vector spaces of polynomials or matrices.
Readiness Assurance.
Before beginning this chapter, you should be able to...
State the definition of a spanning set and determine if a set of Euclidean vectors spans \(\IR^n\text{.}\)
State the definition of linear independence and determine if a set of Euclidean vectors is linearly dependent or independent.
State the definition of a basis and determine if a set of Euclidean vectors is a basis.
Find a basis of the solution space to a homogeneous system of linear equations.
