Chapter 2 · Annotated slides
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.
- 22 slides
- 0 asked in real exams
- 1 related to real exam questions
- 2 practice questions
Slide text
Study notes
Highlight: key words
Tips, tricks & traps
Seen again elsewhere
How it can come up
Exam weight
●●● High
-
A subspace H of ℝⁿ contains 0 and is closed under addition and
scalar multiplication.
Exam weight ●○○ Low
-
Col A: span of A’s columns (subspace of ℝᵐ). Nul A: all
solutions of Ax = 0 (subspace of ℝⁿ).
Exam weight
●●○ Medium
-
dim Nul A = number of free variables; dim Col A = rank =
number of pivot columns.
Exam weight
●●● High
-
Basis of Col A: the pivot columns of the ORIGINAL A (not of
its echelon form). Basis of Nul A: the vectors in the
parametric solution of Ax = 0.
Exam weight
●●● High
-
Rank Theorem: rank A + dim Nul A = n (number of columns).
No slides match this filter.
