And I showed in that video that the span of any set of vectors is a valid subspace. And it's equal to the span of some set of vectors. That is, for X,Y V and c R, we have X + Y V and cX V. And this is a subspace and we learned all about subspaces in the last video. 2 To show that a set is not a subspace of a vector space, provide a speci c example showing that at least one of the axioms a, b or c (from the de nition of a subspace) is violated. We call these the trivial subspaces of V. A subset V of Rn is called a linear subspace of Rn if V contains the zero vector O, and is closed under vector addition and scaling. 1 To show that H is a subspace of a vector space, use Theorem 1. Example Note that V and f0gare subspaces of any vector space V. A linear transformation is injective if and only if its kernel is the trivial subspace f0g. Our rst main result along these lines is the following. 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. De nition (Subspace) A subset W of a vector space V is called a subspace of V if W is a vector space in its own right under the operations obtained by restricting the operations of V to W. However, for linear transformations of vector spaces, there are enough extra constraints to make determining these properties straightforward. 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.
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. 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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