CS223 · Annotated SlidesChapter 4 — Vector Spaces, Subspaces, Bases and Coordinates

Chapter 4 · Annotated slides

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.

29 slides
0 asked in real exams
1 related to real exam questions
0 practice questions

Slide text Study notes Highlight: key words Tips, tricks & traps Seen again elsewhere How it can come up

Slide 1: CS 223: Computational Linear Algebra

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Span{v₁, …, vₚ} is always a subspace.

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

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

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

Exam trapRow space basis comes from the ECHELON form; column space basis from the ORIGINAL A.

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Basis of Row Space

Steps:

Find the echelon form
Indicate the nonzero rows
Take the corresponding rows from the echelon form

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

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Slide 27: Practice problem 1

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Slide 28: Practice problem 2

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