October 27

Announcements

• Webassign 3.3, 4.1, 4.2 due next Thursday
• Sign sheet if you did worksheet 4, there’s a good chance a problem similiar to a problem on worksheet 4 will appear on midterm
• Worksheet 5 posted this weekend
• Watch videos 8,9 of 3blue1brown, determinants are mentioned. Think about it as signed volume.

3.3 Inverses

Theorem: Let A and B be invertible matrices and C and D be matrices. Then

• A^− 1 is also invertible.
• AB is invertible. The inverse is given by (AB)^− 1 = B^− 1A^− 1.
• If AC = AD then C = D.
• If CA = DA then C = D.

Theorem: Let A be a n × n matrix. Let S be the columns of A. Let T(x) = Ax. Then the following are equivalent:

• S spans ℝⁿ
• S is linearly indepedent
• Ax = b has a unique solution for all b ∈ ℝⁿ given by x = A^− 1b.
• T is onto
• T is one-to-one
• T is invertible
• A is invertible

The inverse of [a, b; c, d] is [d,  − b;  − c, a]/det.

Example

Solve the linear system 3x₁ + x₂ = 3 and x₁ − x₂ = 4.

4.1 Subspaces

Defintion: A subset S of ℝⁿ is a subspace if S satisfies the following 3 properties

• S contains 0.
• (closed under addition) If u and v are in S then so is u + v.
• (closed under multiplciation) If r ∈ ℝ and u ∈ S, then ru ∈ S.

Nonexamples:

• If b ≠ 0, then Ax = b is never a subspace.
• The graph y = x² is not a subspace.

Example:

• The span of any set of vectors are a subspace.
• The solutions to Ax = 0 is a subspace.

Consider the matrix A = [3,  − 1, 7,  − 6; 4,  − 1, 9,  − 7;  − 2, 1,  − 5, 5]. The general solution to Ax = 0 is x = s₁( − 2, 1, 1, 0) + s₂(1,  − 3, 0, 1) So the set of solutions is the span of ( − 2, 1, 1, 0) and (1,  − 3, 0, 1).

Definition: The set of solutions to Ax = 0 is called the nullspace of A and is denote null(A).

Definition: Let T : ℝ^m → ℝⁿ be a linear transformation. Then the set {T(x) : x ∈ ℝ^m} is called the range of T. This is a subspace of the codomain. If T is associated to a matrix A, then the range is the span of the columns of A.

The set {x ∈ ℝ^m : T(x) = 0} is called the kernel of T. THis is a subspace of the domain.

• A linear transform is onto if it’s range is equal to the codomain.
• A linear transform is one-to-one if it’s kernel contains only the zero vector.