CS223 · Chapter 2 · Annotated SlidesSubspaces of ℝⁿ
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Subspaces of ℝⁿ

Subspaces, the column space and null space of a matrix, bases, finding bases for Col A (pivot columns of A) and Nul A (parametric solution of Ax = 0), coordinates, dimension and rank, and the Rank Theorem.

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All 22 slides
  1. 1 Subspaces of R
  2. 2 Slide 2
  3. 3 Example A line through the origin is another example of a subspace
  4. 4 Slide 4
  5. 5 Slide 5
  6. 6 In other words The column space is the same as the span of the columns of the matrix
  7. 7 Two conditions must be satisfied in the set of vectors
  8. 8 Slide 8
  9. 9 Example
  10. 10 Slide 10
  11. 11 Slide 11
  12. 12 Slide 12
  13. 13 Slide 13
  14. 14 Slide 14
  15. 15 To find the basis of a To find the basis of the null column space of matrix A : space of matrix A, 1. Find the equivalent 1. Solve the homogenous matrix in Echelon Form equation A x = 0 or Reduced Echelon 2. Write the solution in
  16. 16 Slide 16
  17. 17 Slide 17
  18. 18 Slide 18
  19. 19 Slide 19
  20. 20 Slide 20
  21. 21 Slide 21
  22. 22 Slide 22
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Subspaces of R

Slide 1: Subspaces of R

Notes

  1. A subspace H of ℝⁿ contains 0 and is closed under addition and scalar multiplication.

Exam angle

PredictionQ1

Which is a subspace of ℝ²?

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Example A line through the origin is another example of a subspace

Slide 3: Example A line through the origin is another example of a subspace
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In other words The column space is the same as the span of the columns of the matrix

Slide 6: In other words The column space is the same as the span of the columns of the matrix

Notes

  1. Col A: span of A’s columns (subspace of ℝᵐ). Nul A: all solutions of Ax = 0 (subspace of ℝⁿ).

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Two conditions must be satisfied in the set of vectors

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Example

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Notes

  1. dim Nul A = number of free variables; dim Col A = rank = number of pivot columns.

Exam angle

Past exam · Major 2 (251)

Related question: this topic is examined

Past exam · Final (232)

Related question: this topic is examined

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To find the basis of a To find the basis of the null column space of matrix A : space of matrix A, 1. Find the equivalent 1. Solve the homogenous matrix in Echelon Form equation A x = 0 or Reduced Echelon 2. Write the solution in

Slide 15: To find the basis of a To find the basis of the null column space of matrix A : space of matrix A, 1. Find the equivalent 1. Solve the homogenous matrix in Echelon Form equation A x = 0 or Reduced Echelon 2. Write the solution in

Notes

  1. Trap Take the pivot columns from A itself, not from the reduced matrix.

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Notes

  1. Rank Theorem: rank A + dim Nul A = n (number of columns).

Exam angle

PredictionQ1

A 5×7 matrix has rank 4. dim Nul A =

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