As in Section 6.2.1 on the topic of subspaces I again consulted three textbooks and a set of course notes to see how the concept of linear independence was defined and introduced to the reader. With reference to these sources, textbook authors gave a (virtually) identical account of the concept of linear (in)dependence. Linear (in)dependence is defined in terms of a linear combination so that without first defining a linear combination the definition of linear (in)dependence does not make sense. There was little variation
in the approach taken by the four different authors. For three of the four authors the sequence for introducing the concept of linear (in)dependence consisted of (first) defining a linear combination, followed by span and/or spanning set and then the definition of linear (in)dependence. Greub (1967) defined span (using the to some degree outdated terminology of a “generator”) after he defined linear independence. I state here the definition of a linear combination as given by Poole.
Definition. A vector v is a linear combination of vectors v1, v2, ..., vk if there
are scalars c1, c2, ..., ck such that v = c1v1+ c2v2+ ... + ckvk. (Poole, 2006, p. 12)
Authors differed in defining linear independence first or linear dependence. Although mathematically the definitions are equivalent, the precise formulation of the definitions (the syntax and the wording of the statements) is different. Based on my reading I reproduce here a definition of linear dependence and a definition of linear independence. The versions I give (Definition (1) and Definition (2)) are based on my reading of the three textbooks and the set of course notes.
Definition (1) of linear dependence. Let V be a vector space over K. A subset U of V is linearly dependent if there exist distinct vectors x1, x2, x3, ..., xnin U and
scalars λ1, λ2, . . . , λn in K, not all of which are 0, such that
λ1x1+ λ2x2+ · · · + λnxn= 0.
A set which is not linearly dependent is called linearly independent. (see Hoffman & Kunze, 1961, p. 40)
Alternatively, authors gave the definition for linear independence.
Definition (2) of linear independence. Let V be a vector space over K, U a non-empty subset of V with n elements. Then U is linearly independent if and only if the following condition holds: If
λ1x1+ λ2x2+ · · · + λnxn= 0
where x1, x2, x3, ..., xn are all the elements of U and distinct, and λ1, λ2, . . . , λn∈
K, then
λ1= λ2 = · · · = λk = 0.
A set which is not linearly independent is called linearly dependent. (see Sproston, 1995, p. 6)
Both are “formal” definitions. They use mathematical notation which is deeply em- bedded in mathematical conventions (for expressing definitions, theorems and other
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mathematical statements).5 The definition of linear independence includes the equation λ1x1+ λ2x2+ · · · + λnxn= 0 (λi6= 0)
which expresses a linear relation6 between the vectors xi for i = 1, 2, ..., n. By re-
arranging the equation each vector xi can be expressed as a linear combination of the
other vectors. For example,
x1= − λ2 λ1 x2− λ3 λ1 x3− · · · − λn λ1 xn (λi 6= 0)
where I chose x1 to denote the left-hand side. Choosing each xi in turn by taking
i = 2, 3, ...n, results in linear combinations that are similar to the one I stated above.
Definition (1) states that for a linearly dependent set of vectors it must be possible to write one or more of the vectors as a linear combination of one or more of the remaining vectors. This implies that not all values of λi can be equal to 0. Definition (2) states
that for a linearly independent set this is not possible, that is, none of the vectors xi
can be written as a linear combination of one or more of the remaining vectors. This means also that the equation λ1x1+ λ2x2+ · · · + λnxn= 0 cannot be solved, unless all
the values λi are 0.
All the authors that I consulted had defined the concept of a linear combination prior to introducing the definition of linear independence. Both Poole (2006) and Hoffman and Kunze (1961) stated the definition of linear (in)dependence as “definitions”, Greub (1967) attached no label, while Sproston (1995) attached the label “proposition” (which required a proof).
5In the definition above, the statement
If λ1x1+ λ2x2+ · · · + λnxn= 0,
. . . , then λ1= λ2= · · · = λk= 0,
means that only the trivial solution (that is, the solution λ = 0) exists for solving λ1x1+ λ2x2+ · · · + λnxn= 0.
The definition also contains the clause “if and only if” in its statement of linear independence, namely, U is linearly independent if and only if the following condition holds: . . .
Hence it is possible to define linear independence as follows. If only the trivial solution exists for solving
λ1x1+ λ2x2+ · · · + λnxn= 0,
then the set of vectors U is linearly independent.
6For example, the equations v
3= v1+ v2and v1+ v2− v3= 0 can be referred to as linear relations.
They express a linear relationship between the vectors v1, v2 and v3. In contrast, a linear combination
Sproston (1995) and Poole (2006) gave less formal definitions alongside the formal defi- nitions. Sproston (1995) first wrote,
Let V be a vector space over K, and U a non-empty subset of V . If U has two or more elements, we say U is a linearly independent set if no element of U is a linear combination of other elements of U (if U has only one element x , we say that U is linearly independent if x 6= 0).
and followed with Definition (2). He referred to Definition (2) as a ‘criterion’ for linear independence, and wrote “which is in practice what one checks”, and “in many books, indeed, this is given as the definition of linear independence” (Sproston, 1995, p. 5, italics in original ).
Poole (2006), on the other hand, approached linear (in)dependence by stating Definition (1) first and followed it with a theorem that said,7
A set of vectors . . . is linearly dependent if and only if at least one of the vectors can be expressed as a linear combination of the others. (Poole, 2006, p. 448)
In the approach taken by Poole, the mathematical content was structured in such a way that the concept of linear independence was introduced three times. The first time the author introduced linear independence in the context of the vector space Rn with n-dimensional vectors (often represented as column vectors) as elements of the set; he used geometry and ‘pictures’ as visual aids in relation to vectors being independent in R3; he introduced further theorems to relate linear independence to solution sets and in
terms of the columns of a matrix A. The second time he introduced linear independence in the context of matrices and matrix algebra and referring to row and column space, and the third time in the context of an abstract vector space V . Thus this textbook author presented multiple ways of looking at linear independence which, as I will show, was not unlike the way the lecturer in my study presented this topic.
All authors presented the concept of linear (in)dependence in the same ‘DTP’ (definition- theorem-proof) style as they had done in relation to the concept of a subspace (see Section 6.2.1).
In all textbooks the concept of linear independence was part of a chapter on vector spaces. All the important concepts in linear algebra were introduced in that chapter: linear combination, linear independence, span and spanning sets, basis and change of basis, rank and nullity. As I have shown the concept of linear (in)dependence is defined in terms of a linear relation. In a similar way the concept of span is defined in terms of a
7
I noted that Poole (2006) should have written “if at least one of the vectors can be expressed as a linear combination of one or more of the others”. This was possibly an error not discovered at the proof-reading stage of publication.
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linear combination, and the concept of a basis in terms of a span and linear independence. Dorier (2000) pointed out the interdependence of linear algebra concepts and reported on his research into students’ difficulties with these concepts.
In the next section I discuss how the lecturer in my study designed his teaching of the concept of linear independence, and compare his approach with the one taken by the textbook authors.
6.3.2 The lecturer’s mathematical treatment of linear independence in