Worksheet 2

Due 10/13

1. We know how to obtain the general solution from a linear system. Let’s try to reverse it. Find a linear system who’s general solution is
   (x₁, x₂, x₃, x₄) = (1, 2, 3, 4) + s₁(5, 6, 7, 8) + s₂(9, 0, 1, 2).
   

2. Suppose A is a matrix. Let v, w be distinct (meaning x ≠ y) vectors that solve Ax = 0 so Av = 0 and Aw = 0 (0 here of course means the zero vector!). Let L be the line that passes through v and w. If u is on L, then Au = 0. Why? This exercise suggests that solution spaces are convex.

3. Let z₁, z₂ ∈ ℝ and let S = {(1, z₁, z₂), (2, 1, 0), (1, 0,  − 1)}.

   • Find some values for z₁ and z₂ such that S spans ℝ³.
   • Find some values for z₁ and z₂ such that S does not span ℝ³.
   • Find all values for z₁ and z₂ such that S spans ℝ³. (In the process of solving this problem, some of you will be tempted to divide by zero. Resist that temptation.)

4. Consider the following linear system that came from the book and the lecture. Using row reduction, we see that a general solution is of the form x = s₁(3, 1, 0, 0) + s₂( − 2, 0, 4, 1). Let v₁ = (2, 1,  − 1), v₂ = ( − 6,  − 3, 3), v₃ = ( − 1,  − 1, 2), v₄ = (8, 6, 2).

   • Is {v₁, v₂, v₃, v₄} is linearly independent set? The answer should be no.
   • Express v₁ as a linear combination of v₂, v₃, v₄.
   • Express v₂ as a linear combination of v₁, v₃, v₄.
   • Express v₃ as a linear combination of v₁, v₂, v₄.
   • Express v₄ as a linear combination of v₁, v₂, v₃.

5. Suppose {v₁, v₂, v₃} is a linearly dependent set. Is it always the case that we can write v₁ as a linear combination of v₂ and v₃? If not, come up with a counterexample.

6. Come up with a inconsistent linear system whose associated homogenous linear system is consistent.