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Zeki K.
Math Major from BInghamton University
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Linear Algebra
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Question:

This exercise is supposed to help you understand how the Span Preservation General Lemma works: a) Show that the set S = { [1,0,1], [1,1,3],[0,1,0],[2,-1,0],[-2,0,1] } spans R^3. b) Find a linear dependence among the members of S. c)Use your linear dependence to express one member of S as a linear combination of the others

Zeki K.

(a) The matrix M whose columns are the elements of S has rank 3 ( row reduce to echelon form) so Mx = b has a solution for every b in R^3. (b) There are five columns in M so there are two free columns. There are thus non-trivial solutions to Mx = 0. Pick one by setting at least one of the free variable to something non-zero (such as 1). (c) Solve the linear dependence for one of the vectors with a non-zero coefficient.

Calculus
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Question:

Explain why f(x) = x^5 + 2x has an inverse function.

Zeki K.

Since f(x) = x^5 + 2x, f′(x) = 5x^4 + 2 ≥ 2 > 0 since 5x^4 ≥ 0, so f(x) is an increasing function, which is then one-to-one, so it has an inverse. If we look at the graph online we can also see that it passes the horizontal line test

Economics
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Question:

True or false, and explain: Suppose that good x is a normal good. If a consumer’s income increases, then that will cause a movement to a new point on her ordinary demand curve, with a lower px and a higher quantity of x

Zeki K.

False. It was specified that income changed, not px. Rather than moving to a new point on her ordinary demand curve, the increase in income will shift the demand curve. There will be a horizontal movement from one ordinary demand curve (corresponding to the original income) to another ordinary demand curve (corresponding to the higher income), keeping px constant. Because the good is normal, the demand curve will shift to the right.

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