Overview of Week 6
Math 308M, October 28, 2013
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Reading Assignment:
- Read Section 3.1 - 3.2 for Friday, Oct 18
- Read Section 3.3 for Monday Oct 21.
- Read Section 3.4 for Wednesday Oct 23.
- Read Section 3.5 for Friday Oct 25.
- Read Section 3.6 for Monday Oct 28.
- Read Section 3.7 for Friday Nov 1.
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Homework Assignment:
- Due Oct 23
- Sec. 1.9: 3, 7, 36, 51, 70, 73
10, 16, 17, 22, 34, 40, 53, 69, 79
- Sec. 3.1: 12, 13, 15, 22, 23, 27, 28, 29
- Due Oct 30
- Sec. 3.2: 1, 2, 4, 8, 15, 16, 20, 28
- Sec. 3.3: 1, 4, 6, 10, 14, 20(a, b, c), 26, 28, 33, 35, 36, 38(i, ii), 42, 50
- Sec. 3.4: 2, 6, 10(a, b), 12, 16, 22, 24, 25, 30, 31, 37
- Due Nov 4
- Sec. 3.5: 2, 4, 6, 8, 12, 15, 18, 22, 24, 26, 32, 34, 36, 40
Vocabulary List:
- Section 1.7:
- linear combination of vectors
- zero vector and zero matrix
- linear independence of vectors
- linear dependence of vectors
- standard unit vectors
- nonsingular and singular square matrices
- Section 1.9:
- invertible matrices
- inverse of a matrix
- Steps for computing the inverse of a matrix by hand (page 97)
- the formula for inverse of a 2 by 2 matrix (page 98)
- properties of the inverse (Theorem 17 page 99)
- 5 equivalences for nonsingular matrices (Theorem 18 page 101)
- Section 3.2:
- Vector space properties (Theorem 1 page 168)
- verification procedure for showing that a set W is a subspace (page 171, also Theorem 2 page 169)
- Section 3.3:
- the linear span of a collection of vectors
- the column space of a matrix
- the row space of a matrix
- row equivalence of matrices
- the null-space of a matrix
- the subspace perpendicular to a collection of vectors
- the range of a matrix
- Be able to show that the linear span of a collection of vectors, the null-space of a matrix,
and the range of a matrix are subspaces.
- Section 3.4:
- the natural basis
- spanning sets and minimal spanning sets
- bases
- spanning sets and minimal spanning sets for a subspace
- basis for a subspace
- Section 3.5:
- dimension
- dimension of a suspace
- properties of $p$-dimensional subspaces (Theorem 9 page 207)
- rank of a matrix
- relationship between row and column space dimensions
- Section 3.6:
- orthogonal vectors, set, and bases
- orthonormal basis
- coordinates in a basis
- Gram-Schmidt orthogonalization process
- the Cauchy-Schwartz inequality
- the triangle inequality
- Section 3.7:
- linear tranformation
- the identity, dialations, and contractions
- the matrix of a linear tranformation
- the null space and range of a linear transformation
- rank and nullity
- the rank plus nullity theorem
- orthogonal linear transformations
- rotations and reflections
Key Concepts:
- Section 1.7:
- linear combinations and linear independence
- nonsingular and singular matrices
- Section 1.9:
- the matrix inverse and its relationship to equation solving
- how the inverse in computed
- all of the equivalences to nonsingularity
- Section 3.2:
- subspaces and subspace checking
- Section 3.3:
- special subspaces: linear span, range, null space, perpendicular spaces
- Section 3.4:
- spanning sets, minimal spanning sets, and bases
- Section 3.5:
- Section 3.6:
- orthogonality and orthogonal bases
- the Gram-Schmidt orthogonalization process
- Section 3.7:
- the matrix of a linear tranformation
- orthogonal linear transformations
Skills to Master:
- computing the inverse of a matrix
- checking if a set is a subspace
- representing the range and null space
- computing a minimal spanning set
- computing a basis
- computing and orthonormal spanning set and an orthonormal basis
- computing the matrix associated with a linear transformation
Quiz:
Friday, November 1.
- The first question on the quiz will be based on te vocabulary words from
Sections 1.7, 1.9, 3.2, 3.3, and 3.4. The seccond question will be based on the
homework from these sections.