Subspace definition in linear algebra2/21/2023 However, note that while u = (1, 1, 1) and v = (2, 4, 8) are both in B, their sum, (3, 5, 9), clearly is not. In order for a sub set of R 3 to be a sub space of R 3, both closure properties (1) and (2) must be satisfied. Įxample 2: Is the following set a subspace of R 3? Therefore, the set A is not closed under addition, so A cannot be a subspace. In order for a vector v = ( v 1, v 2 to be in A, the second component ( v 2) must be 1 more than three times the first component ( v 1). For instance, both u = (1, 4) and v = (2, 7) are in A, but their sum, u + v = (3, 11), is not. In the present case, it is very easy to find such a counterexample. If a counterexample to even one of these properties can be found, then the set is not a subspace. To establish that A is a subspace of R 2, it must be shown that A is closed under addition and scalar multiplication. The set V = is a Euclidean vector space, a subspace of R 2.Įxample 1: Is the following set a subspace of R 2? The sum of any two elements in V is an element of V.Įvery scalar multiple of an element in V is an element of V.Īny subset of R nthat satisfies these two properties-with the usual operations of addition and scalar multiplication-is called a subspace of R nor a Euclidean vector space. Thus, the elements in V enjoy the following two properties: The set V is therefore said to be closed under scalar multiplication. In fact, every scalar multiple of any vector in V is itself an element of V. Next, consider a scalar multiple of u, say, The set V is therefore said to be closed under addition. In fact, it can be easily shown that the sum of any two vectors in V will produce a vector that again lies in V. Is also a vector in V, because its second component is three times the first. Now, choose any two vectors from V, say, u = (1, 3) and v = (‐2, ‐6). We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner.The endpoints of all such vectors lie on the line y = 3 x in the x‐y plane. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate "nested subspace" conception that R is a subspace of R for k less than n. 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. 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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