Could someone help to explain this definition with a numerical example? I do not understand it. I am new learner to this topic. thank you.
2026-04-08 22:45:27.1775688327
Could someone explain below definition with example?
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For example, in $\mathbb R^2$ let $U$ be the subspace spanned by the vector $\pmatrix{1\cr 2}$. The orthogonal projection $P_U$ has matrix $$ \pmatrix{1/5 & 2/5\cr 2/5 & 4/5} $$ We have $$\eqalign{\|P_U e_1\|^2 &= \left\|\pmatrix{1/5\cr 2/5}\right\|^2 = (1/5)^2 + (2/5)^2 = 1/5 \cr \|P_U e_2 \|^2 &= \left\| \pmatrix{2/5 \cr 4/5}\right\|^2 = (2/5)^2 + (4/5)^2 = 4/5 \cr}$$ so $\mu(U) = 8/5$.