Problem 4
Prove that for any two scalars \(g\) and \(k\) (a) \(k(A+B)=k A+k B\) \((b)(g+k) A=g A+k A\) (Note: To prove a result, you cannot use specific examples.)
Problem 4
Given the following four matrices, test whether any one of them is the inverse of another: \\[ D=\left[\begin{array}{rr} 1 & 12 \\ 0 & 3 \end{array}\right] \quad E=\left[\begin{array}{rr} 1 & 1 \\ 6 & 8 \end{array}\right] \quad F=\left[\begin{array}{rr} 1 & -4 \\ 0 & \frac{1}{3} \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & -\frac{1}{2} \\ -3 & \frac{1}{2} \end{array}\right] \\]
Problem 5
Given \(u=\left[\begin{array}{l}5 \\ 1\end{array}\right]\) and \(v=\left[\begin{array}{l}0 \\ 3\end{array}\right],\) find the following graphically: \((a) 2 v\) (c) \(u-v\) \((e) 2 u+3 v\) (b) \(u-v\) \((d) v-u\) (f) \(4 u-2 v\)
Problem 5
For \((a)\) through \((d)\) find \(C=A B\) \((a) A=\left[\begin{array}{rr}12 & 14 \\ 20 & 5\end{array}\right] \quad B=\left[\begin{array}{ll}3 & 9 \\ 0 & 2\end{array}\right]\) (b) \(A=\left[\begin{array}{ll}4 & 7 \\ 9 & 1\end{array}\right] \quad B=\left[\begin{array}{lll}3 & 8 & 5 \\ 2 & 6 & 7\end{array}\right]\) (c) \(A=\left[\begin{array}{rr}7 & 11 \\ 2 & 9 \\ 10 & 6\end{array}\right] \quad B=\left[\begin{array}{rrr}12 & 4 & 5 \\ 3 & 6 & 1\end{array}\right]\) \((d) A=\left[\begin{array}{lll}6 & 2 & 5 \\ 7 & 9 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}10 & 1 \\ 11 & 3 \\ 2 & 9\end{array}\right]\) (e) Find (i) \(C=A B,\) and (ii) \(D=B A\), if \(A=\left[\begin{array}{r}-2 \\ 4 \\ 7\end{array}\right] \quad B=\left[\begin{array}{lll}3 & 6 & -2\end{array}\right]\)
Problem 6
Prove that \((A+B)(C+D)=A C+A D+B C+B D\)
Problem 6
Expand the following summation expressions \((a) \sum_{i=2}^{5} x_{i}\) (b) \(\sum_{i=5}^{8} a_{i} x_{i}\) \((c) \sum_{i=1}^{4} b x_{i}\) \((d) \sum_{i=1}^{n} a_{i} x^{i-1}\) \((e) \sum_{i=0}^{3}(x+i)^{2}\)
Problem 7
Rewrite the following in \(\sum\) notation: (a) \(x_{1}\left(x_{1}-1\right)+2 x_{2}\left(x_{2}-1\right)+3 x_{3}\left(x_{3}-1\right)\) (b) \(a_{2}\left(x_{3}+2\right)+a_{3}\left(x_{4}+3\right)+a_{4}\left(x_{5}+4\right)\) (c) \(\frac{1}{x}+\frac{1}{x^{2}}+\dots+\frac{1}{x^{n}} \quad(x \neq 0)\) (d) \(1+\frac{1}{x}+\frac{1}{x^{2}}+\cdots-\frac{1}{x^{n}} \quad(x \neq 0)\)
Problem 7
In the three-dimensional Euclidean space, what is the distance between the following points? (a) (3,2,8) and (0,-1,5) (b) (9,0,4) and (2,0,-4)
Problem 8
The triangular inequality is written with the weak inequality sign \(\leq\), rather than the strict inequality sign \(<.\) Under what circumstances would the \(^{\prime \prime}=^{\prime \prime}\) part of the inequality apply?
Problem 8
Name some situations or contexts where the notion of a weighted or unweighted sum of squares may be relevant.
