Class Outlines

Wed 11/1 Class

Dimension

• Theorem: Any two bases of a subspace W have the same dimension.
• Definition: Dimension of W as the number of vectors in a basis

Read Section 3.5, pp. 202-207, for more details and a careful statement of this -- plus important examples.

Orthogonality

• The dot product of A and B equals 0 if and only if A and B are orthogonal, i.e., when angle AOB = 90 degrees. This is a consequence of the Pythagorean theorem.
• The Law of Cosines follows from the distributive law for dot product.
• Example: Find components of a vector [1 0 2 0]T parallel to and orthogonal to the subspace of solutions of x1 + 2x2 + x3 + 2x4 = 0. (to be continued Friday).

Read 2.3, pp. 138-141 but notice all the reasoning works in n-space as well. You can begin reading 3.6, but it starts off a bit differently.

Fri 11/2 Class

Dimension

• Definition: Rank of a matrix. Nullity of a matrix.
• Theorem: For an m x n matrix A, rank(A) + nullity(A) = n.
• Theorem: For an m x n matrix A, dimension range of A equals dimension of row space of A equals rank.
• Examples

Read the rest of 3.5.

Orthogonality

• Conclusion of Example: Find components of a vector [1 0 2 0]T parallel to and orthogonal to the subspace of solutions of x1 + 2x2 + x3 + 2x4 = 0. (to be continued Friday).
• Finding orthogonal components, projections, coordinates.

Read Section 3.6, pp. 214-218

Study Problems: Section 3.5: #5, 9, 11, 13, 15

Mon 11/5 Class

Orthogonality

• Constructing orthonormal bases by Gram-Schmidt

Read the rest of Section 3.6.

Linear Transformations

• Definition and Examples

Read the first few pages of Section 3.7

Study Problems

• Section 3.5 #17, 19, 27, 21, 23, 33
• Section 3.6 #1-11 odd, 13

Assignment 5 to Turn In (due Wed 11/8)

· Problem 5.1 Let W be the set of vectors x in R⁴ that are solutions to the equation

x1 + x2 + x3 + x4 = 0. Let z = [1, 2, 3, 4]^T. Find vectors u and v so that z = u + v, where u is in W and v is orthogonal to all vectors in W.

^· Problem 5.2. Let S be the span of the vectors s1 = [1, 0, 1, 1] and s2 = [0, 1, 1, 1]. Let w = [1, 0, 0, 0]. Find a vector t so that t is in S and w – t is orthogonal to both s1 and s2.

· Section 3.5: # 24, 26

· Section 3.6: 8, 14, 20