![]() We also present results regarding the coordination between students' concept image and how they interpret the formal definition, situations in which students recognized a need for the formal definition, and qualities of subspace that students noted were consequences of the formal definition. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. We used the analytical tools of concept image and concept definition of Tall and Vinner (Educational Studies in Mathematics, 12(2): 151-169, 1981) in order to highlight this distinction in student responses. A subspace (or linear subspace) of R2 is a set of two-dimensional vectors within R2, where the set meets three specific conditions: 1) The set includes the zero vector, 2) The set is closed under scalar multiplication, and 3) The set is closed under addition. Heres the official definition: 'a spacial continuum with significantly differernt properties from our own.' Oh, you say, eyes glazing over. If you want to be more concise, you can say that a basis of a vector space is a linearly independet spanning. Therefore, I would keep just the first (it is the shortest one) and the second one. 1,318 Your first and third conditions assert the same thing. ![]() This is consistent with literature in other mathematical content domains that indicates that a learner's primary understanding of a concept is not necessarily informed by that concept's formal definition. is a vector space, using the same definition of addition and scalar multiplication as V, then U is called a subspace of V. linear-algebra vector-spaces definition hamel-basis. In interviews conducted with eight undergraduates, we found students' initial descriptions of subspace often varied substantially from the language of the concept's formal definition, which is very algebraic in nature. This paper reports on a study investigating students' ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace.
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