Session

Date

Read & Study Sections

Discussion Topics

Homework Problems

Course Information/Technology Tips

Section 7.1 # 45, 46, 49, 51

Section 4.6 Review the ways of writing a system of linear equations using matrices on page 54. In particular, note that a system has a solution only if the constant matrix b can be written as a linear combination of the column vectors of the coefficient matrix A. This means b must be in the span of the column vectors of A, which is called the column space of A. See the definition of row space and column space of a matrix A on page 226. Note that if A is mxn, then the row space of A is a subspace of n-space, and the column space of A is a subspace of m-space. Study the properties of a row space: (1) row-equivalent matrices have the equal row spaces; (2) if a matrix is reduced to row-echelon form, the nonzero rows of the row-echelon matrix form a basis for the row space; (3) the row space and column space of a matrix have the same dimension; this dimension is called the rank of A. Study section 4.6.

Section 4.6 # 9 - 37 odd, 43, 49, 51

4.6

Section 4.6 What do vector spaces have to do with systems of linear equations?

Section 4.6 # 1 - 8 all

4.5

Section 4.4 How do you test for linear independence and linear dependence? See page 208. A finite set of vectors is linearly dependent if and only if at least one of the vectors is a linear combination of the others. In particular, if one vector is a scalar multiple of another vector, then the set of vectors is linearly dependent. See pages 211 - 212. 

Section 4.5 A set of vectors S={u₁, u₂, ..., u_k } is called a basis for a vector space V if (1) span(S)=V and (2) S is linearly independent. What is the standard basis for 2-space? 3-space? ... n-space? See page 215.

4.5

Section 4.5. A basis S for a vector space V must have enough vectors to span V but not so many that one of them could be written as a linear combination of the other vectors in S. See page 215.

Section 4.5 # 1 - 57 odd

4.4

Section 4.3 A nonempty subset W of a vector space V is a subspace if for any vectors u and v in W, the sum u+v is in W, and if for every scalar c, the product cu is in W. Keep reviewing the fundamentals of vector spaces and subspaces.

Section 4.4 Given a set S of vectors u₁, u₂, ..., u_k of a vector space V,  we say a vector v is a linear combination of the u's if we can find scalars c₁, c₂, ..., c_k so that v=c₁u₁+c₂u₂+...+c_ku_k. The set of all vectors v that can be written as a linear of the u's in set S is called the spanning set of S. Note that the spanning set of S={(1,0), (0,1)} in 2-space is all of 2-space. Study the examples in section 4.4.

4.3

Section 4.3 Some important vector spaces are contained within larger vector spaces. A subspace is a vector space that is a subset of another vector space. Study the definition on page 193. Note theorem 4.5, test for a subspace, actually has three requirements: 0) W is nonempty; 1) W is closed under vector addition; 2) W is closed under scalar multiplication. Study the examples in section 4.3. Note the geometric properties of subspaces of 2-space on page 198 and the geometric properties of subspaces of 3-space on page 199.

Section 4.3 # 1 - 21 odd, 25, 29, 31

4.1

4.2

Section 4.1 The ten properties of n-space described in theorem 4.2, page 180, are the basis for the definition of a vector space.

Section 4.2 Study the definition of vector space on page 186. It is important to realize that a vector space consists of four entities: a set of vectors, a set of scalars, vector addition and scalar multiplication. Unless stated otherwise, you should assume that the set of scalars is the set of real numbers. Study examples 1, 2, 3, 4 and 5 for examples of vector spaces - describe the four entities for each example. Note how versatile the concept of a vector space is: a vector can be a real number, an n-tuple, a matrix, a polynomial, etc. To show that a given set with given operations is not a vector space, you need only find one of the ten axioms in the definition of vector space that is not satisfied. In examples 6, 7 and 8, sets are given that are not vector spaces; name an axiom that is false for each example. 

Section 4.2 # 1 - 25 odd, 31, 33

4.1

Section 4.1 For each integer n>1, Rⁿ denotes n-space: the set of all ordered n-tuples. Each element in n-space can be viewed as a point in Rⁿ or as a vector. The standard operations in Rⁿ of vector addition and scalar multiplication are defined component-wise: see page 179. There are ten mathematical properties that are true for n-space for every positive integer n; see theorem 4.2, page 180. Study these properties closely - these ten properties are the defining properties of the abstraction of Rⁿ which is called a vector space. Study the examples in section 4.1.

Section 4.1

# 1 - 55 odd

Section 3.5 # 1 - 7 odd, 17 - 31 odd, 43, 45, 63, 65

3.2

Section 3.2 Finding the determinant of a triangular matrix is easy: just multiply the entries on the main diagonal. If a given matrix is reduced to a triangular matrix using elementary row operations, what effect do these operations have on the determinant of the matrix? Study theorem 3.3 (elementary row operations and determinants) on page 130. In example 2, page 131, the determinant is evaluated using elementary row operations. Note the conditions that result in a zero determinant given in theorem 3.4 on page 133.

Section 3.2 # 1 - 45 EOO

Section 3.3 # 1 - 29 odd, 37, 45, 49, 51, 57

3.1

Section 2.3 Note in theorem 2.11 that if the coefficient matrix to a system of linear equations is invertible, then the system has a unique solution, x=A^-1b.

Section 3.1 The determinant of a 2x2 matrix is defined on page 120. To find the determinant of matrices of higher order, we can use minors and cofactors. Study the definitions on page 121. Note that by definition, the determinant of a square matrix is found by expanding by cofactors in the first row, page 122, but in fact, the determinant can be found by expanding by cofactors by any row or column (theorem 3.1). The determinant of a triangular matrix is easy to find: multiply the entries on the main diagonal (theorem 3.2).

The university is closed.

The university is closed.

The university is closed.

Section 2.2 A square matrix of order n that has 1's on the main diagonal and 0's elsewhere is called an identity matrix of order n and is denoted by I_n. Identity matrices serve as the identity for matrix multiplication. Note that repeated multiplication can be done for square matrices, so that it makes sense to work with powers of a square matrix. See page 65. The proof that a system of linear equations can have exactly one solution, no solutions or infinitely many solutions is on page 66! The transpose of a matrix is formed by writing its columns as rows. The properties of transposes are on page 67.

And see below.

Section 2.2 Properties of matrix operations: (1) addition is commutative; (2) addition is associative; (3) scalar 1 is a multiplicative identity; (4) multiplication distributes over addition - several cases to consider c(A+B), (c+d)A, A(B+C), (A+B)C; (5) all zero matrices are additive identities for matrices of the same size; (6) sum of a matrix and its additive inverse is a zero matrix; (7) if a scalar product, cA, gives a zero matrix, then the scalar c is zero or the matrix A is a zero matrix; (8) multiplication is not commutative in general; (9) multiplication is associative - two cases to consider A(BC), c(AB). Study and become familiar with the properties of matrix operations in section 2.2.

And see below.

2.1

2.2

Section 1.3 A network consists of junctions/nodes and branches. It is assumed that the total flow into a node equals the total flow out. Hence each node leads to a linear equation, and by solving the system for all the nodes, you can analyze the flow through the network. Study examples 5 - 7.

Section 2.1 Note in example 4 that in general AB does not equal BA for matrices.

Section 2.2 Matrices together with the operations of addition and multiplication have many of the same properties as the real numbers do: addition is commutative, addition is associative, multiplication distributes over addition, the zero matrix is an additive identity, etc. Study the theorems in section 2.2.

2.1

Section 2.1 What two requirements are necessary for equality of matrices? Is matrix addition possible for any two matrices? Describe the process of matrix addition, when it is defined. Is matrix multiplication possible for any two matrices? Describe the processes of scalar multiplication and matrix multiplication, when it is defined. Study the examples in section 2.1.

Section 2.1 # 1 - 35 odd, 37, 39, 41, 49, 51

1.2

Section 1.2 Explain the meaning: mxn matrix; entry a_ij, a square matrix, coefficient matrix of a system, augmented matrix of a system. Describe the Elementary Row Operations. Describe the properties of a matrix written in row-echelon form. Study the method of using Gaussian Elimination with Back-Substitution on page 18. Study Example 6, when using Gaussian elimination to solve a system, what should you be alert for to signal that the system has no solution?