Consider the vectors

$$ a=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], b=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right] $$

For real-valued scalar variable $x$, the value of

$$ \min _x\|a x-b\|_2 $$

is___________(rounded off to two decimal places).

$\|\cdot\|_2$ denotes the Euclidean norm, i.e., for $y=\left[\begin{array}{l}y_1 \\ y_2\end{array}\right],\|y\|_2=\sqrt{y_1^2+y_2^2}$.

Consider the matrix $\begin{bmatrix}1 & k \\ 2 & 1\end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

Let $$x$$ be an $$n \times 1$$ real column vector with length $$l = \sqrt {{x^T}x} $$. The trace of the matrix $$P = x{x^T}$$ is

The state equation of a second order system is

$$x(t) = Ax(t),\,\,\,\,x(0)$$ is the initial condition.

Suppose $$\lambda_1$$ and $$\lambda_2$$ are two distinct eigenvalues of A and $$v_1$$ and $$v_2$$ are the corresponding eigenvectors. For constants $$\alpha_1$$ and $$\alpha_2$$, the solution, $$x(t)$$, of the state equation is