October 9

Announcements

• Section 2.1, 2.2 due this Thursday
• Section 2.3, 3.1 due next Thursday
• Midterm next Wednesday in class
  • 1.1 - 3.1 (maybe 3.2)
• Worksheet 1 solutions has been posted
• Worksheet 2 has been posted, due this Friday
• Wednesday office hours to be held in classroom?

Homogenous Systems

Let A be a matrix. Then A(x + y) = Ax + Ax and A(x − y) = Ax − Ay.

Example: Find a general solution for the linear system $ $ Using row reduction, we see that a general solution is of the form x = (1, 0,  − 5, 0) + s₁(3, 1, 0, 0) + s₂( − 2, 0, 4, 1).

The solution to the homogenous system is x = s₁(3, 1, 0, 0) + s₂( − 2, 0, 4, 1).

Let x_p be a particular solution Ax = b. Then solutions have the form x_g = x_p + x_h, where x_p is a particular solution and x_h is the general solution to the homogenous equations.

Linear Indepedence and Span

Theorem: Let A = [a_i] and b be a vector in ℝⁿ. Then the following are equivalent (if one is true then they are all true, if one is false then they are all false).

• The set {a₁, …, a_m} are linearly independent.
• The vector equation x₁a₁ + x₂a₂ + … + x_ma_m = b has at most one solution.
• The linear system [a₁ a₂ … a_m|b] has at most one solution.
• The equation Ax = b has at most 1 solution.

Example: Consider the vectors a₁ = (1, 7,  − 2), a₂ = (3, 0, 1), and a₃ = (5, 2, 6). Set A = [a_i]. Show that the columns of A are linearly independent and that Ax = b has a unique solution for every b in ℝ³.

Example: Let u₁ = (1,  − 1, 2), u₂ = (2,  − 1, 2), u₃ = ( − 2, 5,  − 10), u₄ = (3,  − 4, 8). The associated matrix has reduced echelon form:
$$ \begin{bmatrix} 1 & 2 & -2 & 3 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$
Is {u₁, …, u₄} linearly independent? Can we write u₁ as a linear combination of u₂, …, u₄?

If a set of vectors is not linearly indepedent, can every vector be written as a linear combination of the other vectors? In other words, is every vector in the span of the other vectors?

Section 3.1 Linear Transformations

We can write linear equations as Ax = b. We can think of it as A sending x to b.

Definition: A function T : ℝ^m → ℝⁿ is a linear transformation if for all vectors u, v ∈ ℝ^m and all scalars r, we have

• T(u+v) = T(u) + T(v)
• T(ru) = rT(u).

Examples:

• What are some examples of functions that aren’t linear transforms? quadratic, ax+b
• Consider the function given by T(x₁, x₂) = (3x₁ − x₂, 2x₁ + 5x₂). What is T(1, 2)? Show that this is a linear transformation. Is it associated to a matrix?
• Projections are linear transforms.
• Let A be some matrix. Then T(x) = Ax is a linear transform. Make up some example in class.

A matrix, A, is said to be an n × m matrix if it has n rows and m columns. If m = n, then A is a sqaure matrix.

Theorem: Let A be an n × m matrix, and define T(x) = Ax. Then T : ℝ^m → ℝⁿ is a linear transform. Moreover, all linear transform are of this form.

Example: Consider the linear transform with matrix
$$ A= \begin{bmatrix} 1 & -2 & 4 \\ 3 & 0 & 5 \end{bmatrix}. $$
Is (3, 4) in the range of A?

Theorem: Let A = [a₁ a₂ … a_m be a n × m matrix, and let T : ℝ^m → ℝⁿ with T(x) = Ax be a linear transformation. Then

• A vector w is in the range of T if and only if Ax = w is a consistent linear system.
• The range of T is the span of the columns (this is also called the column space).

If time, talk about 1-1 and onto