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. SUBSPACE IN LINEAR ALGEBRA: INVESTIGATING STUDENTS’ CONCEPT IMAGES AND INTERACTIONS WITH THE FORMAL DEFINITION Megan Wawro George Sweeney Jeffrey M. If x 1 and x 2 are in N (A), then, by definition, A x 1 0 and A x 2 0. To prove that N (A) is a subspace of R n, closure under both addition and scalar multiplication must be established. The first thing we have to do in order to comprehend the concepts of subspaces in linear algebra is to completely understand the. Vector SubspaceHow to prove set of all even function is a subspace Set of all function such that it satisfies the given D.E is a subspaceHello friends,Wel. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. This subset actually forms a subspace of R n, called the nullspace of the matrix A and denoted N (A). If S is empty then by definition its span is the trivial subspace. 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. 2.15 Lemma In a vector space, the span of any subset is a subspace.
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. Lets begin with the definition vector space and subspace and then the theorem about. 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. used ( ) to illustration the proof of the theorems.