Research Scholarships Screening Test Saturday, January 19, 2019 Q. 1.9

Question

\( n \in \mathbb{N} \), \( n \geq 2 \).  
\( M_n(\mathbb{R}) \) (respectively, \( M_n(\mathbb{C}) \)) will denote the set of all \( n \times n \) matrices with entries from \( \mathbb{R} \) and is identified with \( \mathbb{R}^{n^2} \) when considered as a topological space.  
\( A \) be an \( m \times n \) matrix with real entries.

Let \( \mathbf{x}_0 \in \mathbb{R}^3 \) be the column vector such that \( \mathbf{x}_0^\top = (1, 1, 1) \). Let  
\[
V = \{ A \in M_3(\mathbb{R}) \mid A \mathbf{x}_0 = \mathbf{0} \}.
\]  
What is the dimension of \( V \)?

Comments

what is $A$ and $M_3$?

 

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added. please check

Answer 1

A * (1,1,1) = (0,0,0)

This means : 

1. The sum of Row 1 must be 0.
2. The sum of Row 2 must be 0.
3. The sum of Row 3 must be 0.

Final Dimension = (Total Freedom) - (Number of Rules)
Final Dimension = 9 - 3 = 6