(Solved): The experiment involves rolling the die a fixed number of times (3 million times), which constitutes ...

Consider the vectors ${\mathbf{v}}_1=\left(\begin{array}{l}4\\ 6\\ 0\end{array}\right)\text{ and }{\mathbf{v}}_2=\left(\begin{array}{c}-2\\ 7\\ 1\end{array}\right)$ in ${\mathbb{R}}^3$. (a) Select in the list below ALL the corrects proofs that $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is a linearly independent set. Multiple selection warning: In a multiple selection question, one point is awarded for each correct selection, and one point is deducted for each incorrect selection (but you cannot get a negative mark for the question in the end). The vectors ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ are not parallel, therefore $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is a linearly independent set. The vectors ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ are elements of ${\mathbb{R}}^3$, which is a vector space of dimension 3 . Since the number of vectors in the set $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is less than the dimension of the ambient vector space (since $2<3$ ), we must have that $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is a linearly independent set. If ${\lambda }_1={\lambda }_2=0$, then ${\lambda }_1{\mathbf{v}}_1+{\lambda }_2{\mathbf{v}}_2=0$, and therefore $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is a linearly independent set. The only solution of the equation ${\lambda }_1{\mathbf{v}}_1+{\lambda }_2{\mathbf{v}}_2=0$ is ${\lambda }_1={\lambda }_2=0$, therefore $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is a linearly independent set. The scalar product ${\mathbf{v}}_1\cdot {\mathbf{v}}_2\ne 0$, therefore $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ is a linearly independent set. (b) In the following three questions you are asked to find a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has a certain dimension. In each case, either enter a suitable vector in the box, using Maple notation, or enter the word IMPOSSIBLE in capital letters if you believe that there is no vector ${\mathbf{v}}_3$ which does what is asked. Syntax advice. Vectors must be entered in Maple notation. For example, the vector $\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right)$ is entered as $⟨1,2,3⟩$ You can use the preview button next to any entry field to make sure that your syntax is correct. - Find (if possible) a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 1. Enter the word impossible if there is no such vector. Answer: - Find (if possible) a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 2 . Enter the word impossible if there is no such vector. Answer. - Find (if possible) a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 3 . Enter the word impossible if there is no such vector. Answer. (c) Suppose that you are doing lots of different examples of the final part of question (b). You do not want to be always considering many different vectors: so you decide to choose a set $S$ of vectors such that at least one of the vectors in $S$ will be a correct answer to the question, no matter what two vectors are given. Syntax advice. Vectors must be entered in Maple notation. For example, the vector $\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right)$ is entered as $⟨1,2,3⟩$ You can use the preview button next to any entry field to make sure that your syntax is correct. - Find (if possible) a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 1 . Enter the word impossible if there is no such vector. Answer: 固 굑. - Find (if possible) a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 2 . Enter the word impossible if there is no such vector. Answer. 圆 교․ - Find (if possible) a vector ${\mathbf{v}}_3$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 3 . Enter the word impossible if there is no such vector. Answer. (c) Suppose that you are doing lots of different examples of the final part of question (b). You do not want to be always considering many different vectors: so you decide to choose a set $S$ of vectors such that at least one of the vectors in $S$ will be a correct answer to the question, no matter what two vectors are given. In other words: you want a set $S$ of vectors in ${\mathbb{R}}^3$ such that if any two linearly independent vectors ${\mathbf{v}}_1,{\mathbf{v}}_2$ in ${\mathbb{R}}^3$ are given, there will be a vector ${\mathbf{v}}_3$ in your set $S$ such that $\mathrm{span}\left({\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right)$ has dimension 3 . - What is the minimum number of vectors you can take in the set $S$ ? Enter your answer as a nonnegative integer. Answer. - Give an example of a set $S$, of the size you just stated, which will do what is wanted. Syntax advice. For example, the set of vectors $\left\{\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right),\left(\begin{array}{l}4\\ 5\\ 6\end{array}\right)\right\}$ is entered as $\{⟨1,2,3⟩,⟨4,5,6⟩\}$. Note the curly brackets which indicate a set. Recall that the order of the vectors in a set does not matter. You can and should use the preview button next to any entry field to make sure that your syntax is correct Answer: 固 요.