# Dealing with the Abstraction of Vector Space Concepts

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## Abstract

University mathematics students often find the content of linear algebra difficult because of the abstract and highly theoretical nature of the subject as well as the formal logic required to carry out proofs. This chapter explores some specific difficulties experienced by students when negotiating vector space and subspace concepts. Seventy-three in-service mathematics teachers’ responses to two items testing the ability to prove that a given set is not a subspace and that a given set is a subspace of a vector space were studied in detail. Follow-up interviews on the written work were conducted to identify the participants’ ways of understanding. The action–process–object–schema (APOS) theory was used to unpack the structure of the concepts. Findings reveal that the teachers struggled with the vector sub-space concepts mainly because of prior non-encapsulation of prerequisite concepts of sets and binary operations and difficulties with understanding the role of counter-examples in showing that a set is not a vector subspace.

## Keywords

APOS Vector subspace Binary operations Counter-example Vector space## References

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