Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Linear Algebra Inner Product Spaces, Orthonormal Basis
Let P be a 2 × 2 real orthogonal matrix and \(\vec x\) is a real vector [x1, x2]T with length
\(\|\vec{x}\|=\left(x_1^2+x_2^2\right)^{\frac{1}{2}}\). Then which one of the following statements is correct?
1
\(\|P \vec{x}\| \leq\|\vec{x}\|\) where at least one vector satisfies \(\|P \vec{x}\|<\|\vec{x}\|\)
2
\(\|P \vec{x}\| =\|\vec{x}\|\) for all vector \(\vec x\)
3
\(\|P \vec{x}\| \ge \|\vec{x}\|\) where at least one vector satisfies \(\|P \vec{x}\| >\|\vec{x}\|\)
4
No relationship can be established between \( \|\vec{x}\|\) and \(\|P \vec{x}\|\)