Some issues about the introduction of first concepts in linear algebra during tutorial sessions at the beginning of university

Abstract

Certain mathematical concepts were not introduced to solve a specific open problem but rather to solve different problems with the same tools in an economic formal way or to unify several approaches: such concepts, as some of those of linear algebra, are presumably difficult to introduce to students as they are potentially interwoven with many types of difficulties as formal ones and far away from the actual knowledge of the students. The purpose of this paper is to propose a methodology for studying the introduction of such concepts in linear algebra during tutorial sessions at the beginning of university, the wording of the concepts being yet presented during lectures. For this purpose, we amend a general methodology of Pariès, Robert and Rogalski inside the general framework of Activity Theory. This methodology lets us take into account several specificities of these concepts and studies the mathematical activity the teacher organises for students and the way he manages the relationship between students’ actual activities and mathematical tasks. We also present an implementation of this methodology based on a French university course to illustrate our approach and discuss its possibilities.

Appendix

1.1 Sheet 2 : Vector spaces and subspaces

Exercise 1

In ℝ², define the operations (x, y) + (x′, y′) = (x + x′, y + y′), α.(x, y) = (α x, 0). Is ℝ² endowed with these operations a vector space?

Exercise 2

Show that the set of matrices $M=\left (\begin {array}{cc} a & c\\ b & d\end {array}\right )$ such that a + d = 0 is a vector space.

Exercise 4

Let a and b be real numbers. Show that the set of solutions of the differential equation y″ + a y′ + b y = 0 over [0; 1] is a vector space.

Exercise 5

Let 𝓟 denote the set of functions of class C ^∞ from ℝ to ℝ. Which of the following subspaces are vector spaces and which are not?

• (1) $\lbrace (x_{1},x_{2})\in \mathbb {R}^{2} \ | \ x_{1} = 5x_{2}\rbrace$.

• (2) $\lbrace (x_{1},x_{2})\in \mathbb {R}^{2} \ | \ x_{1} +x_{2}=7 \rbrace$.

• (3) $\lbrace f\in \mathcal {P} \ | \ f(0) = 1\rbrace$.

• (4) $\lbrace f\in \mathcal {P} \ | \ f(x) = f(-x) \textrm { for all }x\in \mathbb {R}\rbrace$.

• (5) $\lbrace (x_{1},x_{2},x_{3})\in \mathbb {R}^{3} \ | \ x_{1}+x_{2} = 0 \textrm { or } x_{1}=x_{3}\rbrace$.

• (6) $\lbrace (x_{1},x_{2},x_{3})\in \mathbb {R}^{3} \ | \ {x_{1}}^{2}+{x_{2}}^{2} = {x_{3}}^{2}\rbrace$.

New exercise

Let F (resp. F _a ) be the vector subspace of ℝ² spanned by the vector (1, 1) (resp. (2, a)), where a is a real parameter. Determine the vector subspace of ℝ² spanned by F∪F _a according to a.

Exercise 6

Let E be the set of functions from ℝ to ℝ, F (resp. G) be the subset of E that consists of even (resp. odd) functions. Show that F and G are vector subspaces of E and that E = F ⊕ G.

Exercise 7

Define the following vector subspaces of ℝ²:

$$D_{1}=\lbrace (a,b)\in{\mathbb{R}}^{2} \ | \ b=0\rbrace; \ D_{2}=\lbrace (a,b)\in{\mathbb{R}}^{2} \ | \ a=0\rbrace; \;D_{3}=\lbrace (a,b)\in{\mathbb{R}}^{2} \ | \ a=b\rbrace.$$

Show that ℝ² = D ₁ + D ₂ + D ₃. Is it a direct sum?

Exercise 8

Let D be the vector subspace of ℝ³ spanned by the vector (−1, 1, 2) and define the set P = {(x, y, z) ∈ ℝ³ | x + 2y − z = 0}. Show that P is a vector subspace strictly included in ℝ³. Show that any vector u = (x, y, z) ∈ ℝ³ can be uniquely decomposed as (x, y, z) = (x′, y′, z′)+(x″, y″, z″), (x′, y′, z′) ∈ P and (x″, y″, z″) ∈ D.

Exercise 9

Consider the n-tuples $(x_{1},\cdots , x_{n}), \ (x^{\prime }_{1},\cdots , x^{\prime }_{n})$ and let λ be a real paramater. Express the following n-tuples as a liear combination of (x ₁,⋯ , x _n ) and $(x^{\prime }_{1},\cdots , x^{\prime }_{n})$ : $(x_{1}+x^{\prime }_{1},\cdots , x_{n}+x^{\prime }_{n})$, $(x_{1}-x^{\prime }_{1},\cdots , x_{n}-x^{\prime }_{n})$, (0,⋯ ,0), (x ₁,⋯ , x _n ), (λ x ₁,⋯ , λ x _n ).

Exercise 10

Can we find two real numbers x and y such that the vector v = (−2, x, y,3) is a linear combination of the vectors u ₁ = (1, −1, 1, 2) and u ₂ = (−1, 2, 3, 1)?

Exercise 11

Give a cartesian equation of the following vector subspaces:

• (1) F = Vect{(1, 3)} ⊂ ℝ².

• (2) G = Vect{(1, 2, 3); (1, 0, 1)} ⊂ ℝ³.

• (3) H = Vect{(1, 0, 1, 0)} ⊂ ℝ⁴.

Exercise 12

Define

$$E=\left\{ \left(\begin{array}{ccc} a-b & b-c & 2c\\ 2a & a+b & -b\\ b & c & a\end{array}\right), \ a,b,c\in \mathbb{R}\right\}.$$

Show that E is a vector subspace of ℳ₃(ℝ). Show that E is spanned by the following matrices

$$\left(\begin{array}{ccc} 1 & 0 & 0\\ 2 & 1 & 0\\ 0 & 0 & 1\end{array}\right), \ \ \left(\begin{array}{ccc} -1 & 1 & 0\\ 0 & 1 & -1\\ 1 & 0 & 0\end{array}\right), \ \ \left(\begin{array}{ccc} 0 & -1 & 2\\ 0 & 0 & 0\\ 0 & 1 & 0\end{array}\right).$$

1.2 Sheet 3 : Sets of vectors

Exercise 1 (the only part of question (4) that has been videorecorded concerns the question of the linear dependence of the considered subset)

In the vector space ℝ³, decide whether or not the subsets consisting of the following vectors are linearly dependent or span ℝ³.

• (1) (1)u ₁ = (−1, 2, 1), $u_{2} = (-\frac {1}{\sqrt {2}}, \sqrt {2}, \frac {1}{\sqrt {2}})$.

• (2) u ₁ = (−1, 1, 1), u ₂ = (1, 1, −2), u ₃ = (1, 2, 1).

• (3) u ₁ = (1, 2, 3), u ₂ = (3, 2, 1), u ₃ = (1, 1, 1).

• (4) u ₁ = (1, 1, 1), u ₂ = (m, 1, m), u ₃ = (m, m, m ²), where m is a real parameter.

Exercise 8

Let u = (1, 2, 3), v = (2, −1, 1), $w=(2,-3,-1)\in \mathbb {R}^{3}$.

• (1) Show that the subset consisting of u , v and w is linearly dependent and give a basis of F = Vect{u, v, w}.

• (2) Give a cartesian equation of F.

Exercise 5

Define v ₁ = (2, 1, 3, 1), v ₂ = (1, 2, 0, 1), $v_{3}=(-1,1,-3,0) \in \mathbb {R}^{4}$ and let E = Vect{v ₁, v ₂, v ₃}. Give the dimension and a basis of E.