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# MTH 4100: Midterm 1 Practice 1

Problem 1

Solve the system of equations \begin{eqnarray*} 5x-2y+3z+15w&=&1\newline 3x+3y-z+2w&=&2\newline 11x+10y-3z+9w&=&7. \end{eqnarray*}

Problem 2

Assume that $$A(-3,2)$$, $$B(8,9)$$, and $$C(0,12)$$ are three given points. Let $$X$$ be the midpoint of the segment $$AB$$. Determine the vector $$\overrightarrow {CX}$$.

Problem 3

Determine the real number $$k$$ such that the vectors $$\displaystyle \overrightarrow u=\left[\begin{array}{c}8\newline k\newline -6\end{array}\right]$$ and $$\displaystyle \overrightarrow v=\left[\begin{array}{c}1\newline 2\newline 3\end{array}\right]$$ are orthogonal.

Problem 4

Consider the following five vectors in $$\mathbb R^4$$: $\overrightarrow m=\left[\begin{array}{c} 2\newline 0 \newline 0 \newline 2\end{array}\right], \quad \overrightarrow n= \left[\begin{array}{c} 0\newline 0 \newline 2 \newline 0\end{array}\right], \quad \overrightarrow p=\left[\begin{array}{c} 0\newline 1 \newline 0 \newline -2\end{array}\right], \quad \overrightarrow q=\left[\begin{array}{c} 2\newline -2 \newline 6 \newline 6\end{array}\right], \quad \mbox{and}\quad \overrightarrow r=\left[\begin{array}{c} 4\newline -2 \newline 6 \newline 6\end{array}\right].$

• (a) Determine the real numbers $$\alpha$$, $$\beta$$, and $$\gamma$$ such that $$\overrightarrow q=\alpha \overrightarrow m+\beta\overrightarrow n+\gamma\overrightarrow p$$.

• (b) Does the vector $$\overrightarrow q$$ belong to the vector space $$\displaystyle V=\mbox{span}\left\{\overrightarrow m,\overrightarrow n,\overrightarrow p\right\}$$?

• (c) Does the vector $$\overrightarrow r$$ belong to the vector space $$\displaystyle V=\mbox{span}\left\{\overrightarrow m,\overrightarrow n,\overrightarrow p\right\}$$?

• (d) Determine a basis and the dimension of the vector space $$\displaystyle W=\mbox{span}\left\{\overrightarrow m, \overrightarrow n, \overrightarrow p, \overrightarrow q\right\}$$.

Problem 5

Assume that $$d\geq 3$$ is a positive integer and that the vectors $$\overrightarrow u,\overrightarrow v\in\mathbb R^d$$ are linearly independent. Prove that the vectors $$4\overrightarrow u+3\overrightarrow v$$ and $$2\overrightarrow u-\overrightarrow v$$ are linearly independent as well.

Problem 6

Assume that the sequence $$a_n$$ is defined as $$a_0=2$$, $$a_1=10$$, and $$a_{n+2}=10a_{n+1}-21a_n$$ for every $$n\geq 0$$. Use the principle of mathematical induction to prove that $$a_n=3^n+7^n$$ for all $$n\geq 0$$.

Problem 7

Let $$F$$ be the set of all polynomials $$P$$ of degree $$3$$ or less for which $$P(7)\cdot P(12)=0$$. Using the set notation $$F$$ can be defined as $F=\left\{ P\in \mathcal P_3: P(7)\cdot P(12)=0\right\}.$ The addition and scalar multiplication in $$F$$ are defined in the same standard way as in $$\mathcal P_3$$. Is $$F$$ a subspace of $$\mathcal P_3$$? Provide a rigorous mathematical justification for your answer.