Why is $E(Y|X)$ is projection of $Y$(so that error is zero) on plane containing all functions of X and passing through origin?

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Why is $E(Y|X)$ is projection of $Y$(so that error is zero) on plane containing all functions of X and passing through origin?

I have recently started studying probability and statistics using harvard lectures. In that professor tried to establish relation between expectation and linear algebra. I did not understand how this holds. kindly elaborate

At 19.46 https://www.youtube.com/watch?v=gjBvCiRt8QA&list=PL2SOU6wwxB0uwwH80KTQ6ht66KWxbzTIo&index=27