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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.