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Tutor profile: Sehkai L.

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Sehkai L.
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Questions

Subject: Linear Algebra

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Question:

Consider $$P_n$$, the set of all polynomials over $$\mathbb{R}$$ of degree less than $$n$$. Recall that the "standard basis" for $$P_n$$ is given by $$ \{1,t,t^2,\dots,t^{n-1}\} $$. Define the linear operator $$ T: P_3 \to P_5$$ by $$ Tf(x) = \int\limits_0^x\int\limits_0^t p(t) \; dt $$. Find the matrix representation for $$T$$, with respect to the standard bases of $$P_3$$ and $$P_5$$. What is the kernel of $$T$$? The range of $$T$$?

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Sehkai L.
Answer:

$$T$$ is represented by the matrix $$\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{6} & 0 \\ 0 & 0 & \frac{1}{12} \end{pmatrix} $$. The kernel of $$T$$ is $$\{0\}$$, the zero subspace (so in particular $$T$$ is injective). The range of $$T$$ is $$\text{span}\{t^3, t^4, t^5 \}$$.

Subject: Set Theory

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Question:

Let $$a_n$$, $$n=1,2,3,\dots$$, be a sequence of real numbers. For any subset $$ A \subseteq \mathbb{N}$$, define $$ \sum\limits_{n \in A} a_n $$ to be the sum of all numbers $$a_n$$ whose index $$n$$ belongs to the set $$A$$. For example, if $$A = \{1,3,6,10 \}$$, then $$\sum\limits_{n\in\mathbb{A}} a_n = a_1 + a_3 + a_6 + a_{10} $$. If $$\varnothing$$ denotes the empty set, then what is $$\sum\limits_{n\in \varnothing} a_n$$? What about $$ \prod\limits_{n\in\varnothing} a_n $$ (the product instead of the sum)?

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Sehkai L.
Answer:

$$\sum\limits_{n\in\varnothing} a_n = 0$$ and $$\prod\limits_{n\in\varnothing} a_n = 1 $$.

Subject: Calculus

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Question:

Give an example of a point $$a$$ and a differentiable, non-constant function $$f$$ whose derivative $$f'$$ satisfies $$f'(a) = 0 $$ but $$f$$ has neither a local max nor a local min at $$a$$.

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Sehkai L.
Answer:

Any odd monic polynomial will do! For example, consider $$f(x) = x^5$$ and the point $$a = 0$$. Then $$f'(x) = 5x^4$$ so $$f'(0) = 0$$, but $$f(x) = x^5$$ does $$\textit{not}$$ have a local max or min at $$0$$. This example is helpful in remembering that it is not enough for a function's derivative to equal 0 for the original function to have an extremum. Not only must the derivative must be 0, the derivative must be changing signs as well!

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