(1 point) Check the true statements below: A. For each y and each subspace W, th
ID: 3115074 • Letter: #
Question
(1 point) Check the true statements below: A. For each y and each subspace W, the vector y projw (v is orthogonal to w. B. In R3, the orthogonal complement of the xy-plane is the yz-plane. C. If { e. e2, es) is an orthonormal basis of V and W span(el, e3), then w1span(e) y itself. depend on the orthogonal basis for W used to compute projwv). D. If y 1s in a subspace W, then the orthogonal projection projwof y onto W is E. itzis orthogonal to ui. and 112 and if W = span { ui, u), then z must be in WL . F. The orthogonal projection projw(of y into a subspace W can sometimesExplanation / Answer
A. The statement is True. The vector y- projW(y) is orthogonal to W.
B. The statement is False. In R3, the orthogonal complement of xy plane is the z-Axis.
C. The statement is True. If {e1,e2,e3} is an orthonormal basis for V and W = span {e1,e3}, then W consists of all the vectors orthogonal to both e1 and e3 and hence W = span{e2} as e2 is orthogonal to both e1 and e3.
D. The staement is True. y is a linear combination of the vectors in the basis of W.
E. The staement is True. If W = span{u1,u2}, then W consists of all the vectors orthogonal to both u1 and u2. Since z is orthogonal to both u1 and u2, hence z is in W.
F. The statement is False. The orthogonal projection projW (y) of y onto a subspace W is always independent of the basis for W.
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