Overview of Week 5
Math 308M, October 21, 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
Vocabulary List:
- Section 1.6:
- the distributive and associative properties for matrix multiplication and addition
- transpose of a matrix
- symmetric matrices
- square matrices
- the identity matrix
- the main diagonal of a matrix
- Euclidean norm (2-norm)
- 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
Key Concepts:
- Section 1.6:
- algebraic properties of matrix algebra
- transpose
- square matrices and special square matrices: identity, zero, symmetric, diagonal
- Euclidean norm
- 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:
Skills to Master:
- vector representation of the solution set
- checking linear independence
- checking singularity and nonsingularity
- generating linear systems for networks (Sec 1.4)
- 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
Quiz:
Friday, October 25.
- The first question of the quiz will ask your to recite, use, or illustrate
one of the vocabulary words from Sections 1.7, 1.9, 3.2 and 3.3 listed above.
The second problem will be computational in nature and will be similar to
the homework problems from sections 1.7, 1.9, and 3.2.