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Section
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Content
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Suggested Problems
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Section 3.1
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- vector space (linear space)
- binary operation addition
- commutative
- associative
- existence of an additive identity (called 0)
- existence of additive inverse for each a (called -a)
- scalar multiplication
- distributive property i
- distributive property ii
- associativity of scalar multiplication
- unity property of scalar multiplication
- Example: Rn
- addition = component-wise addition
- scalar multiplication = component-wise scalar multiplication
- R2, R3
- vectors identified with "directed line segments"
- component-wise addition identified with geometry of parallolegram rule
- scalar multiplication identified with geometric scaling of vector
- length of vector given by Pathagorean rule
- Example: Rnxm
- addition = matrix addition
- scalar multiplication = scalar multiplication on matrices
Example: Pn
- addition = polynomial addition
- scalar multiplicaiton = multiplication of scalars on polynomials
- Example: C[a,b}
- addition = function addition
- scalar multiplicaiton = multiplication of scalars on functions
- Theorem 3.1.1 (Additional Properties of Vector Spaces)
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4, 6, 8, 9, 10, 11, 12
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Section 3.2
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- definition of a subspace
- examples for R2
- examples for R2x2
- examples for P3
- null space of a matrix N(A)
- span of a set of vectors
- Theorem 3.2.1
- spanning set for a vector space V
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1, 2, 3, 5, 7, 9, 10, 11, 14, 17
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Section 3.3
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- linear independence
- linear dependence
- linear independence of two vectors
- Theorem 3.3.1 Linear independence of n vectors in Rn
- Theorem 3.3.2
- Linear Independence Conditions/Criteria
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1, 2, 6, 7, 11
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Section 3.4
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- basis
- Examples
- Theorem 3.4.1
- Corolary 3.4.2
- dimension
- finite dimensional vector space
- infinite dimensional vector space
- Theorem 3.4.3
- Standard Bases
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1, 2, 3, 5, 7, 10, 11, 14
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Section 3.5
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- Rn
- Standard Basis for Rn
- Ordered Basis in Rn
Coordinates w.r.t Ordered Basis
- Problem
- 1. Given coordinates in standard basis find coordinates in ordered basis
- 2. Given coordinates in ordered basis find coordinates in standard basis
- Solution
- Problem 2: Transition Matrix from Ordered Basis to Standard Basis U
- Problem 1: Transition Matrix from Standard Basis to Ordered Basis U-1
- Transition Matrix S from Ordered Basis F to Ordered Basis E
- General Finite Dimensional Vector Space
- Ordered Basis
Coordinates w.r.t Ordered Basis
- Problem
- Given coordinates in ordered basis E find coordinates in ordered basis F
- Solution
- Problem 2: Transition Matrix S from Ordered Basis F to Ordered Basis E
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1, 2, 3, 5, 6, 10
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Section 3.6
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- vector space (linear space)
- Row Space of A
- Column Space of A
- Theorem: Two row equivalent matrices have the same row space
- Theorem 3.6.2 Consistency Theorem for Linear Systems
- Theorem: A (nxn) is non-singular iff the column space of A = Rn
- Rank of A
- Nullity of A
- Theorem 3.6.5. Rank-Nullity Theorem
- Meta-Theorem: Two row equivalent matrices have different column spaces, but the same column dependency relationships
- Theorem 3.6.6 dim(row space of A) = dim(col space of A)
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1, 2, 4, 7, 8, 10, 11, 12
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