Linear algebra questions
Math 220 Written Homework 2 Be sure to show all your work including each step of elimination and clearly indicate your final answer. Only use a calculator or computer algebra system on problems marked with [C]. 1. Determine if each set of vectors is a linearly independent or linearly dependent set. If they are linearly dependent, state the dependence relation (the dependence is relation is in the form c1⃗v1 + c2⃗v2 + c3⃗v3 = 0, where you must find all non-zero c1 , c2 , c3 ). 1 2 −1 (a) , , 0 3 4 1 3 4 0 , 4 , 4 (b) −2 5 3 1 0 1 0 , 1 , 1 (c) 1 0 1 2. Provide a geometric description of the span. In each part, it will either be a point, a line, a plane, or a 3D space. And they will live in either R2 or R3 . So a sample answer would look like a plane in R3 . Justify your response, perhaps begin by removing any redundant vectors. 3 1 1 , 2 (a) Span 1 0 1 3 5 (b) Span , , 1 3 5 1 2 (c) Span , 0 3 0 2 (d) Span , 0 3 1 2 3 1 0 1 1 2 3. Consider the subspace of R4 , W = span 0 , 2 , 2 , 4 . 0 0 0 0 (a) Find a basis for W . (b) Identify the dimension of W based off of what you found in (a). Then give a geometric description of W in the form: living in . W is a 1 1 0 0 4. Recall ⃗e1 = 0 , ⃗e2 = 1 , ⃗e3 = 0We know that {⃗e1 , ⃗e2 , ⃗e3 } forms a basis for R3 . 0 0 1 (a) Come up with a basis {⃗v1 , ⃗v2 , ⃗v3 } for R3 that does not involve ⃗e1 , ⃗e2 , ⃗e3 (try to find something simple). (b) Because {⃗e1 , ⃗e2 , ⃗e3 } is a basis, every vector in R3 can be uniquely written as linear combination of vectors in {⃗e1 , ⃗e2 , ⃗e3 }. For example, 1 2 = ⃗e1 + 2⃗e2 + 3⃗e3 . 3 Find a, b, c so that 1 2 = a⃗v1 + b⃗v2 + c⃗v3 . 3 Show (using techniques from previous weeks) that a, b, c is a unique solution. Congrats, you just found a new coordinate system for R3 ! The coordinates for the vector above is (a, b, c). 5. Compute the following: 1 1 1 (a) 1 0 0 1 1 1 (b) 1 0 1 1 1 2 (c) 1 0 1 2
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