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Section
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Content
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Suggested Problems
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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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Section 4.1
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- Examples
- Properties
- L(0V) = 0W
- L(-x) = -L(x)
- Linear combinations
- Kernel of L
- Image of a Subspace under L
- Range of L
- Theorem 4.1.1
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1, 4, 5, 6, 7, 9, 11, 17, 19
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Section 4.2
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- Theorem 4.2.1 Matrix representation of Linear Transformation from Rn to Rm
- Theorem 4.2.2 Matrix representation of Linear Transformation from V to W
- Theorem 4.3.3 Construction of Matrix A for case of V = Rn and W = Rm in Theorem 4.2.2
- Corollary Above construction via row equivalent transformation of an augmented matrix
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1, 2, 3, 4, 6, 14, 18
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Section 4.3
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- Linear Operator
- Theorem 4.3.1
- Similarity
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1, 3, 4, 5
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