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Calculate each employee’s new salary after the merit increase. (Number only; 1000 or 1,000) Alex: $[AlexMoney] Brianna: $[BriannaMoney] Carlos: $[CarlosMoney]

BrightWave Consulting is reviewing its compensation system for Business Analysts. The salary range for this position is: Minimum: $48,000 Midpoint: $60,000 Maximum: $72,000 Three employees currently earn: Employee Current Salary Performance Rating Alex $54,000 Exceeds Expectations Brianna $60,000 Meets Expectations Carlos $70,000 Exceeds Expectations The company uses the following merit increase grid: Performance Rating Compa-Ratio 80–90% Compa-Ratio 91–110% Compa-Ratio 111–120% Exceeds Expectations 7% 5% 3% Meets Expectations 4% 3% 2% Below Expectations 2% 0% 0%  

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?