Basis and Dimension

Q. The dimension of the null space of the matrix $\left[\begin{array}{ccc}0& 1& 1\\ 1& -1& 0\\ -1& 0& -1\end{array}\right]$ is

1. 0
2. 1
3. 2
4. 3

Q. The number of linearly independent solutions of the system of equations

$\left[\begin{array}{ccc}1& 0& 2\\ 1& -1& 0\\ 2& -2& 0\end{array}\right]\left[\begin{array}{c}{X}_1\\ X_2\\ X_3\end{array}\right]=0$ is equal to

1. 1
2. 2
3. 3
4. 0

Q. It is given that $X_1,X_2,...X_M$ are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors $X_1,X_2,...X_M,-X_1,-X_2,...-X_M$ is

1. 2M
2. M + 1
3. M
4. dependent on the choice of $X_1,X_2,...X_M$

Q. Let A be an $n\times n$ matrix with rank $r(0<r<n$). Then AX=0 has p independent solutions, where p is

1. r
2. n
3. n - r
4. n + r