Q$1$ Answer the following questions with justifications.

a$)$ Given two linearly independent $D1$ vectors $v_1$ and $v_2,$ where $D>3,$ we intend to create a $D3$ matrix where each column of the matrix is a linear combination of the given vectors $v_1$ and $v_2.$ If possible, find vectors $v_1$ and $v_2$ and values for the coefficients in the linear combination such that the resulting matrix has full column rank. Otherwise, explain with a suitable mathematical argument why this is not possible.

b$)$ Consider a $nn$ matrix A which we can write as $A=L^{T}L$ where $L$ is a lower$-$triangular matrix. We have a function $F(x,R,y)$ that returns the value $x^{T}Ry$ where $R$ is an $nn$ matrix and $x$ and $y$ are $n1$ vectors respectively. If we are given both the matrix A and $L,$ what is the smallest number of calls that need to be made to this function in terms of $n$ in order to decide that the given matrix $A$ is positive definite, and why?

c$)$ We are given a function $f(x)=e^{x+}.$ Can the series $1+3x+5x^2$ be the Taylor's polynomial to second degree that approximates this function around $x=0?$