Finite-dimensional vector spaces

Learning tasks

A) Existence of finite bases
I) Vector spaces of finite dimension
1) Recognizing a finite-dimensional space	Ex 1 Ex 2 Ex 3
II) Maximal free family in a finitely generated space
2) Using the max-free bound	Ex 4 Ex 5 Ex 6
III) Theorem of the incomplete basis and theorem of the extracted basis
3) Applying the incomplete-basis algorithm	Ex 7 Ex 8 Ex 9
4) Extracting a basis from a generating family	Ex 10 Ex 11 Ex 12
B) Dimension and rank
I) Dimension of a finite-dimensional vector space
5) Computing the dimension of a space	Ex 13 Ex 14 Ex 15
II) Cardinal of free and generating families in dimension \(n\)
6) Showing a family is a basis by counting	Ex 16 Ex 17 Ex 18
7) Cardinality traps	Ex 19 Ex 20 Ex 21
III) Dimension of a product
8) Computing the dimension of a product	Ex 22 Ex 23 Ex 24
IV) Rank of a finite family of vectors
9) Computing the rank of a family	Ex 25 Ex 26 Ex 27
C) Subspaces and dimension
I) Dimension of a subspace
10) Comparing subspace dimensions	Ex 28 Ex 29 Ex 30
II) Grassmann formula
11) Applying Grassmann	Ex 31 Ex 32 Ex 33 Ex 34
III) Supplementary subspaces in finite dimension
12) Finding a supplement	Ex 35 Ex 36 Ex 37
13) Using the two-of-three characterisation	Ex 38 Ex 39 Ex 40
IV) Bases adapted to a subspace or to a direct sum
14) Completing a basis of \(F\) into a basis of \(E\)	Ex 41 Ex 42 Ex 43
15) Writing a basis adapted to a direct sum	Ex 44 Ex 45 Ex 46
D) Application: dimensions of natural solution spaces	Convention In this section we use the standard results on linear homogeneous differential equations and linear recurrence relations: the solution space of an order-\(r\) homogeneous linear equation is a vector space of dimension \(r\), once existence and uniqueness from initial data are known. These results are admitted here and proved in the chapters Differential equations and Recurrent sequences.
I) First-order linear homogeneous ODE
16) Computing dimensions of solution spaces	Ex 47 Ex 48 Ex 49
II) Second-order linear ODE with constant coefficients
17) Computing dimensions of solution spaces	Ex 50 Ex 51 Ex 52
III) Linear recurrent sequences of order 2
18) Computing dimensions of solution spaces	Ex 53 Ex 54 Ex 55

Lesson
Text book

Exercises Correction

Conventions

Throughout this exercise sheet, unless otherwise stated, \(\mathbb{K}\) denotes either \(\mathbb{R}\) or \(\mathbb{C}\), \(E\) denotes a \(\mathbb{K}\)-vector space, and \(n, p\) are positive integers. Reference vector spaces are \(\mathbb{K}^n\), \(\mathcal{M}_{n,p}(\mathbb{K})\), \(\mathbb{K}[X]\), \(\mathbb{K}_n[X]\). The general definitions of generating family, free family, basis, \(\mathrm{Vect}\), sum, direct sum, supplement were given in the previous chapter Vector spaces; this sheet drills the finite-dimensional theory: existence of a finite basis, the dimension theorem, rank, Grassmann formula, supplements in finite dimension, adapted bases.

A) Existence of finite bases
    I) Vector spaces of finite dimension
      1) Recognizing a finite-dimensional space Ex 1 Ex 2 Ex 3
    II) Maximal free family in a finitely generated space
      2) Using the max-free bound Ex 4 Ex 5 Ex 6
    III) Theorem of the incomplete basis and theorem of the extracted basis
      3) Applying the incomplete-basis algorithm Ex 7 Ex 8 Ex 9
      4) Extracting a basis from a generating family Ex 10 Ex 11 Ex 12
B) Dimension and rank
    I) Dimension of a finite-dimensional vector space
      5) Computing the dimension of a space Ex 13 Ex 14 Ex 15
    II) Cardinal of free and generating families in dimension \(n\)
      6) Showing a family is a basis by counting Ex 16 Ex 17 Ex 18
      7) Cardinality traps Ex 19 Ex 20 Ex 21
    III) Dimension of a product
      8) Computing the dimension of a product Ex 22 Ex 23 Ex 24
    IV) Rank of a finite family of vectors
      9) Computing the rank of a family Ex 25 Ex 26 Ex 27
C) Subspaces and dimension
    I) Dimension of a subspace
      10) Comparing subspace dimensions Ex 28 Ex 29 Ex 30
    II) Grassmann formula
      11) Applying Grassmann Ex 31 Ex 32 Ex 33 Ex 34
    III) Supplementary subspaces in finite dimension
      12) Finding a supplement Ex 35 Ex 36 Ex 37
      13) Using the two-of-three characterisation Ex 38 Ex 39 Ex 40
    IV) Bases adapted to a subspace or to a direct sum
      14) Completing a basis of \(F\) into a basis of \(E\) Ex 41 Ex 42 Ex 43
      15) Writing a basis adapted to a direct sum Ex 44 Ex 45 Ex 46
D) Application: dimensions of natural solution spaces

Convention

In this section we use the standard results on linear homogeneous differential equations and linear recurrence relations: the solution space of an order-\(r\) homogeneous linear equation is a vector space of dimension \(r\), once existence and uniqueness from initial data are known. These results are admitted here and proved in the chapters Differential equations and Recurrent sequences.

    I) First-order linear homogeneous ODE
      16) Computing dimensions of solution spaces Ex 47 Ex 48 Ex 49
    II) Second-order linear ODE with constant coefficients
      17) Computing dimensions of solution spaces Ex 50 Ex 51 Ex 52
    III) Linear recurrent sequences of order 2
      18) Computing dimensions of solution spaces Ex 53 Ex 54 Ex 55

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