CS223 · Chapter 4 · Annotated SlidesVector Spaces, Subspaces, Bases and Coordinates
Original
Overview

Vector Spaces, Subspaces, Bases and Coordinates

Vector space axioms and subspaces, spans as subspaces, null space, column space and row space, kernel and range of a linear transformation, spanning sets and bases, and coordinate vectors.

How to read this page

Every original slide is shown unchanged. Yellow marks on a slide point to a numbered note: hover or tap a mark to jump to its note, and hover a note to see its marks. Questions tagged Past exam appeared on a real paper; Prediction questions are likely angles, not confirmed. Pick an answer to each question: a wrong pick tells you why it is wrong, so you can try again.

All 29 slides
  1. 1 CS 223: Computational Linear Algebra
  2. 2 Slide 2
  3. 3 Slide 3
  4. 4 Slide 4
  5. 5 Slide 5
  6. 6 Slide 6
  7. 7 Slide 7
  8. 8 Slide 8
  9. 9 Slide 9
  10. 10 The set of all linear combinations of the rows of a matrix
  11. 11 Slide 11
  12. 12 Slide 12
  13. 13 Slide 13
  14. 14 Is v in the span?
  15. 15 Slide 15
  16. 16 Slide 16
  17. 17 Slide 17
  18. 18 Slide 18
  19. 19 Slide 19
  20. 20 Slide 20
  21. 21 Basis of Row Space
  22. 22 Slide 22
  23. 23 Slide 23
  24. 24 Slide 24
  25. 25 Slide 25
  26. 26 Slide 26
  27. 27 Practice problem 1
  28. 28 Practice problem 2
  29. 29 Practice problem 3
Slide 1 / 29

CS 223: Computational Linear Algebra

Slide 1: CS 223: Computational Linear Algebra
Slide 2 / 29

Slide 2

Slide 2: Slide 2
Slide 3 / 29

Slide 3

Slide 3: Slide 3
Slide 4 / 29

Slide 4

Slide 4: Slide 4
Slide 5 / 29

Slide 5

Slide 5: Slide 5
Slide 6 / 29

Slide 6

Slide 6: Slide 6
Slide 7 / 29

Slide 7

Slide 7: Slide 7
Slide 8 / 29

Slide 8

Slide 8: Slide 8
Slide 9 / 29

Slide 9

Slide 9: Slide 9
Slide 10 / 29

The set of all linear combinations of the rows of a matrix

Slide 10: The set of all linear combinations of the rows of a matrix

Notes

  1. Span{v₁, …, vₚ} is always a subspace.

Slide 11 / 29

Slide 11

Slide 11: Slide 11
Slide 12 / 29

Slide 12

Slide 12: Slide 12
Slide 13 / 29

Slide 13

Slide 13: Slide 13
Slide 14 / 29

Is v in the span?

Slide 14: Is v in the span?
Slide 15 / 29

Slide 15

Slide 15: Slide 15

Notes

  1. Kernel of T = Nul A; range of T = Col A.

Slide 16 / 29

Slide 16

Slide 16: Slide 16
Slide 17 / 29

Slide 17

Slide 17: Slide 17
Slide 18 / 29

Slide 18

Slide 18: Slide 18

Notes

  1. Spanning Set Theorem: removing a vector that is a combination of the others doesn’t change the span; some subset is a basis.

Slide 19 / 29

Slide 19

Slide 19: Slide 19
Slide 20 / 29

Slide 20

Slide 20: Slide 20

Notes

  1. Basis: a linearly independent spanning set. Col A: pivot columns of A. Row A: nonzero rows of an echelon form.

  2. Trap Row space basis comes from the ECHELON form; column space basis from the ORIGINAL A.

Exam angle

Past exam · Major 2 (251)

Related question: this topic is examined

Past exam · Final (231)

Related question: this topic is examined

Slide 21 / 29

Basis of Row Space

Slide 21: Basis of Row Space
Slide 22 / 29

Slide 22

Slide 22: Slide 22
Slide 23 / 29

Slide 23

Slide 23: Slide 23

Notes

  1. Coordinates: x = c₁b₁ + … + cₙbₙ ⇒ [x]_B = (c₁, …, cₙ); find them by solving P_B·c = x.

Slide 24 / 29

Slide 24

Slide 24: Slide 24
Slide 25 / 29

Slide 25

Slide 25: Slide 25
Slide 26 / 29

Slide 26

Slide 26: Slide 26
Slide 27 / 29

Practice problem 1

Slide 27: Practice problem 1
Slide 28 / 29

Practice problem 2

Slide 28: Practice problem 2
Slide 29 / 29

Practice problem 3

Slide 29: Practice problem 3