Vector spaces

Learning tasks

A) Vector spaces
I) Definition and rules of calculation
1) Recognizing a vector space	Ex 1 Ex 2 Ex 3
II) Reference vector spaces
2) Identifying reference spaces	Ex 4 Ex 5 Ex 6
III) Linear combinations
3) Computing linear combinations	Ex 7 Ex 8 Ex 9
B) Subspaces
I) Definition and characterization
4) Identifying subspaces	Ex 10 Ex 11 Ex 12 Ex 13 Ex 14
II) Reference subspaces
5) Recognizing function subspaces	Ex 15 Ex 16 Ex 17 Ex 18
III) Intersection of subspaces
6) Computing intersections	Ex 19 Ex 20 Ex 21
IV) Subspace spanned by a set
7) Working with Vect	Ex 22 Ex 23 Ex 24 Ex 25
C) Families of vectors
I) Generating families
8) Finding a generating family	Ex 26 Ex 27 Ex 28
II) Free and dependent families
9) Showing a family is free	Ex 29 Ex 30 Ex 31 Ex 32 Ex 33 Ex 34
III) Bases and coordinates
10) Identifying a basis	Ex 35 Ex 36 Ex 37
11) Finding coordinates	Ex 38 Ex 39 Ex 40
D) Sum of subspaces
I) Sum of two subspaces
12) Computing a sum	Ex 41 Ex 42 Ex 43 Ex 44
II) Direct sum
13) Recognizing a direct sum	Ex 45 Ex 46 Ex 47 Ex 48
III) Supplementary subspaces
14) Showing supplementary	Ex 49 Ex 50 Ex 51 Ex 52
15) Finding a supplementary	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. The reference \(\mathbb{K}\)-vector spaces are \(\mathbb{K}^n\), \(M_{n,p}(\mathbb{K})\), \(\mathbb{K}[X]\), \(\mathbb{K}_n[X]\), \(\mathbb{K}^\Omega\) (functions from a set \(\Omega\) to \(\mathbb{K}\)), and \(\mathbb{K}^\mathbb{N}\) (sequences). The notation \(\mathrm{Vect}(X)\) denotes the subspace spanned by a part \(X\) of \(E\). The dimension theory (existence of a finite basis, dimension formula, theorem of the incomplete basis, rank) is deferred to the next chapter, Finite-dimensional vector spaces; this sheet stays in the dimension-free setting.

A) Vector spaces
    I) Definition and rules of calculation
      1) Recognizing a vector space Ex 1 Ex 2 Ex 3
    II) Reference vector spaces
      2) Identifying reference spaces Ex 4 Ex 5 Ex 6
    III) Linear combinations
      3) Computing linear combinations Ex 7 Ex 8 Ex 9
B) Subspaces
    I) Definition and characterization
      4) Identifying subspaces Ex 10 Ex 11 Ex 12 Ex 13 Ex 14
    II) Reference subspaces
      5) Recognizing function subspaces Ex 15 Ex 16 Ex 17 Ex 18
    III) Intersection of subspaces
      6) Computing intersections Ex 19 Ex 20 Ex 21
    IV) Subspace spanned by a set
      7) Working with Vect Ex 22 Ex 23 Ex 24 Ex 25
C) Families of vectors
    I) Generating families
      8) Finding a generating family Ex 26 Ex 27 Ex 28
    II) Free and dependent families
      9) Showing a family is free Ex 29 Ex 30 Ex 31 Ex 32 Ex 33 Ex 34
    III) Bases and coordinates
      10) Identifying a basis Ex 35 Ex 36 Ex 37
      11) Finding coordinates Ex 38 Ex 39 Ex 40
D) Sum of subspaces
    I) Sum of two subspaces
      12) Computing a sum Ex 41 Ex 42 Ex 43 Ex 44
    II) Direct sum
      13) Recognizing a direct sum Ex 45 Ex 46 Ex 47 Ex 48
    III) Supplementary subspaces
      14) Showing supplementary Ex 49 Ex 50 Ex 51 Ex 52
      15) Finding a supplementary Ex 53 Ex 54 Ex 55

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