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Section
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Content
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Suggested Problems
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Section 4.7
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- Coordinate of a vector with respect to a basis
- Coordinate matrix
- Notation [x]B
- Change of basis in Rn
- Transition matrix from basis B' to B
- Inverse of a transition matrix
- Theorem 4.21 Transition matrix from basis B to B'
- Coordinate representation in general vector space V
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Pages 210-211
5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 35, 37
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Section 4.8
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- Linear differential equation
- Wronskian of a set of functions
- Wronskian test for linear independence
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Pages 219-223
13, 14, 17, 20, 21
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Section 5.1
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- Length or norm of a vector in Rn
- Length of a scalar multiple
- Unit vector
- Distance between two vectors in Rn
- Dot product of two vectors in Rn
- Properties of the dot product
- Cauchy-Schwarz inequality
- Angle between two non-zero vectors in Rn via dot product
- Definition of orthogonal
- Triangle Inequality in Rn
- Pythagorean Theorem in Rn
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Pages 235-236
7-8, 11-12, 20-21, 25-26, 41-42, 48-51
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Section 5.2
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- Definition of inner product on a vector space
- Examples of inner products for Rn, Mm,n, C[a,b]
- Properties of inner products
- Definitions of length, distance and angle
- Cauchy-Schwarz inequality
- Triangle Inequality
- Pythagorean Theorem
- Orthogonal Projection of a vector u on to a vector v
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Pages 245-247
17, 19, 21, 23, 29, 35, 39, 41, 47-49, 73, 75, 78-79
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Section 5.3
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- Orthogonal sets
- Orthonormal sets
- Orthogonal basis, Orthonormal basis
- Theorem 5.10 Orthogonal sets is linearly independent
- Corollary to Theorem 5.10, Page 251
- Theorem 5.11 Coordinates of vector with respect to an orthogonal basis
- Theorem 5.12 Gram-Schmidt Orthogonalization Process
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Pages 257-258
1, 4, 7, 10, 13, 21, 25, 27, 31, 35, 39, 41
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Section 5.4
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- Least Square Regression Line
- Least Squares Problem
- Orthogonal Subspaces
- Orthogonal Complement
- Direct Sum
- Properties of Orthogonal Subspaces
- Projection onto a Subspace which is the span of an orthogonal basis
- Theorem 5.15 Orthogonal Projection and Distance
- Fundamental subspaces of a matrix
- Theorem 5.16 Fundamental Subspaces of a Matrix
- Solution of the Least Squares Problem
- Normal equation of the least squares problem
- Orthogonal projections onto a subspace
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Pages 269-270
5-7, 15-17, 19-21, 23-25, 29-31, 33-35
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Section 5.5
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- Least squares approximations in calculus
- Definition of least squares approximation
- Theorem 5.19 Least squares approximating function
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Pages 282-283
63-65, 69-71
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