October 16

Announcements

• Solutions for worksheet 2 and 3 are posted
• Midterm on Wednesday

Review

Talk about some worksheet solutions.

Give an example of each of the following. If it is not possible, write “NOT POSSIBLE”.

• Give an example of a linear system with no solutions
• Give an example of a linear system with infinitely many solutions
• Give an example of 4 vectors in ℝ³ that are linearly independent.
• Give an example of 3 vectors in ℝ³ that are spanning.
• Give an example of a linear transformation T : ℝ² → ℝ² such that T(3, 0) = (2, 3).
• Give an example of a linear transformation T : ℝ² → ℝ³ that is onto.

Let v₁ = (1, 0, 0, 0), v₂ = (1, 2, 0, 0), v₃ = (2, 3, 4, 0), v₄ = (1, 2, 3, 0). Let S = {v₁, v₂, v₃, v₄}.

• Is S a spanning set? If not, what is a vector not in the span?
• Is S a linearly independent set? If not, write one of the vectors as a linear combination of the others.
• Let A = [v₁ v₂ v₃ v₄]. Give a nontrivial solution to Ax = 0.
• Let T be the linear transformation defined by T(x) = Ax. What is the dimension of the domain and codomain of T?
• Is T one-to-one? Why or why not?
• Is T onto? Why or why not?

And then we go over old midterms.

kdev au13 problem 1. kdev au13 problem 2. talk a bit about homogeneous case just keep going, just keep going, just keep going,….