2114-Homework1 (1).pdf

2114-Homework1 (1).pdf

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Math 2114 Carpet Rides Homework 1 1. After some discussion, we can to an agreement in class that (allowing for negatives and non-integer increments), everywhere on the plane could be reached using the hoverboard 3 1 and magic carpet 1 2 (i.e., there was nowhere that Gauss could hide). We learned another way to say this is that Span 3 1 , 2 1 = R 2 . (a) Make a geometric argument supporting this claim. Include a graphical illustration as well as a verbal explanation. (b) Make an algebraic argument supporting this claim using vector notation as linear combinations of vectors. 2. Now supposes your modes of transportations are given by a b and c d , where a, b, c, d are real numbers. What must be true about the relationship(s) amount a , b , c , and d in order to STILL be able to reach any point in R 2 ? Justify your reasoning. 3. Suppose ~v 1 , ~v 2 are vectors in R 2 . Is 0 0 in Span( { ~v 1 ,~v 2 } )? Explain your reasoning in enough detail to justify your conclusion.
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