Question

\text { In this problem, you are given two matrices } A, B \in \mathbb{R}^{2 \times 2} \text { and a vector } x \in \mathbb{R}^{2} A=\left[\begin{array}{ll}

1 & 2 \\

2 & 4

\end{array}\right], \quad B=\left[\begin{array}{ll}

1 & 2 \\

3 & 4

\end{array}\right], \quad x=\left[\begin{array}{l}

2 \\

1

\end{array}\right] and asked to answer the following questions about them. What is A x B? (b) What is Az? What is ? ) What is xx? ) What is the projection of a onto the subspace spanned by the columns of A? ) Let ƒ : R² → R be the function given by f(z) = z¹ Az. What is the gradient of f with respect to z,i.e. Vzf(z)? \text { For the function } f \text { defined above, what is } \nabla_{z}^{2} f(z) \text { (the Hessian of } f \text { with respect to the vector } z \in \mathbb{R}^{2} \text { )? } \text { What is the maximizer of } f \text { among all vectors with unit Euclidean length, }\|z\|_{2}=1 \text { ? }

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