Calcular la proyección del vector \( \vec{u}=(-3,2,1) \) sobre el vector \( \vec{v}=(-1,2,4) \) a. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \) b. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21}, \frac{44}{21}\right) \) c. \( \vec{w}=\left(-\frac{11}{21},-\frac{22}{21}, \frac{44}{21}\right) \) d. \( \vec{w}=\left(\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \)

\( \mathrm{U}=\{1,2, \mathbf{3}, \ldots \mathbf{1 0}\} \) is a given universal set. \( \mathrm{O}=\{\mathbf{1}, \mathbf{3}, \mathbf{5}, \mathbf{7} \), \( \mathrm{P}=\{\mathbf{3}, \mathbf{5}, 7\} \) are any two given sets. (i) Is the set \( O \) a subset of \( U \) ? Give reason.

4. If \( \quad\left[\begin{array}{cccccc}0 & 0 & -3 & -2 & 0 & 5 \\ -2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -2 & 3 & 0 & 0 & 0 \\ -2 & 8 & 3 & 6 & 2 & 7 \\ 0 & 2 & 3 & 0 & 0 & 0\end{array}\right] \operatorname{compute} \operatorname{det}(A),(\operatorname{adj}(A))^{-1} \) and \( \operatorname{det}(\operatorname{adj}(A)) \)

IX. Please determine whether the set of vectors \( \left\{\begin{array}{lll}\alpha_{1} & \alpha_{2} & \alpha_{3}\end{array}\right\} \) is linearly dependent or not, where \( \alpha_{1}=\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)^{T}, \alpha_{2}=\left(\begin{array}{llll}2 & 1 & 1\end{array}\right)^{T}, \alpha_{3}=\left(\begin{array}{lll}3 & 2 & 1\end{array}\right)^{T} \). (Write the answer for the questions on your answer sheet. 10 points)

Q. 15 Find the eigenvalues and eigenvectors of the matrix \( \left[\begin{array}{ll}6 & 5 \\ 3 & 4\end{array}\right] \).

1. Representar en un diagrama de Venn el conjunto \( [A-(A \cap B)]^{C} \). Considere a) \( A \) subconjunto de \( B \) b) \( B \) subconjunto de \( A \)