Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors
$$ \{ [2 \ -3 \ \alpha], \ [3 \ -1 \ 3], \ [1 \ -5 \ 7] \}$$
does not form a basis of $\mathbb{R}^3$ is _______.
Let the sets of eigenvalues and eigenvectors of a matrix B be $$\{ {\lambda _k}|1 \le k \le n\} $$ and $$\{ {v_k}|1 \le k \le n\} $$, respectively. For any invertible matrix P, the sets of eigenvalues and eigenvectors of the matrix A, where $$B = {P^{ - 1}}AP$$, respectively, are
Consider a system of linear equations Ax = b, where
$$A = \left[ {\matrix{ 1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr { - 1} \hfill & {\sqrt 2 } \hfill & { - 3} \hfill \cr } } \right]$$, $$b = \left[ {\matrix{ 1 \cr 3 \cr } } \right]$$
This system is equations admits __________.
vector $$X = \left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$.
The number of distinct real values of k for which the equation AX = 0 has infinitely many solutions is _______.