Author: Anonymous

Baby Yoda vs. Baby GrootThe Galactic Council has gathered to settle the most important debate in the universe:Who wins the Ultimate Cute-Off?Answer the following multiple-choice questions carefully. The fate of the galaxy depends on it. [You will get full points for whatever you choose]

Perform the indicated matrix operations.Note: Your answer will be in terms of the variables \(a\), \(b\), and \(c\).\(\begin{bmatrix} 1 & 2\\ a & -1\end{bmatrix}\begin{bmatrix} 3 & 0\\ 0 & 1\\ 2 & b\end{bmatrix}^T – 2\begin{bmatrix} 1 & 0 & c \\ 0 & 1 & 1\end{bmatrix}=\underline{\hspace{1in}}\)

Let B be the ordered basis for \(\mathbb{R}^3\) given by \(B=\left\{\begin{bmatrix} 1\\ -1\\ 0\end{bmatrix}, \begin{bmatrix} 0\\ 2\\ 1\end{bmatrix}, \begin{bmatrix} -1\\ 0\\ -2\end{bmatrix}\right\}\). Using this ordered basis, apply the Gram-Schmidt orthonormalization process to find a corresponding orthonormal basis for W.

Let W be the subspace of \(\mathbb{R}^5\) spanned by \(B=\left\{\vec{v}_1, \vec{v}_2, \vec{v}_3\right\}=\left\{\begin{bmatrix} 1\\ 0\\ -1\\ 2\\ 3\end{bmatrix}, \begin{bmatrix} 4\\ 2\\ 0\\ -2\\ -5\end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 2\\ 1\\ -1\end{bmatrix}\right\}\).Find a basis for \(W^{\perp}\), the orthogonal complement of W.

Identify which one of the following sets of vectors in \(\mathbb{R}^3\) is linearly independent.

Identify which of the following vectors is orthogonal to \(\vec{u}=(2, 0, 3, 1)\).

An \(n\times n\) matrix \(A\) is diagonalizable if and only if…

Consider the matrix \(A=\begin{bmatrix} -3 & 0 & 20\\ 1 & -3 & 1\\ 0 & 0 & 2\end{bmatrix}\). whose characteristic polynomial is given by \(p(\lambda)=(\lambda+3)^2(\lambda-2).\)Note: You do not need to find this characteristic polynomial. It is given to you above.Answer each of the following:Use this characteristic polynomial to find all eigenvalues of the matrix $A$.Find a basis for the eigenspace corresponding to each of the eigenvalues of $A$, indicating which basis corresponds to each eigenvalue.Is the matrix $A$ diagonalizable?

Imagine a divide the dollar game in which John and Jane are trying to reach a settlement to share $100.  They get one opportunity (that consists of a single proposal by one of them and either acceptance or rejection of that proposal by the other) to reach such a deal.  If they fail to reach a deal, the $100 disappears.  They initially flip a coin to determine who will make the first offer.  If it lands on heads, John can choose to make the first proposal or allow Jane to make the first proposal.  If it lands on tails, Jane can choose to make the first proposal or allow John to make the first proposal.  Which of the following is the most likely equilibrium outcome of this interaction if the coin flip lands on heads and both players are trying to maximize their income?

A preterm infant receiving chronic furosemide therapy develops hypochloremia and metabolic alkalosis. Which pharmacologic effect most likely contributed?