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10 Nov 2019
Consider VR2 with the nonstandard inner product (x, )22 -2yi (a) Prove directly from the axioms that this indeed defines an inner product on V (b) Prove or disprove: the set fe, e2 is an orthonormal basis of V with respect to this nonstandard inner product. If it's not, perform the Gram-Schmidt orthogonalization process to find an orthonormal basis B for V. (c) Express an arbitrary vector x-(x1,x2) E V as a linear combination of basis vectors in an orthonormal basis that you found in (b). (d) Let W Span(-2,3)) c V. For an arbitrary vector x-(x1, r2) E V, determine the projection of x onto W.
Consider VR2 with the nonstandard inner product (x, )22 -2yi (a) Prove directly from the axioms that this indeed defines an inner product on V (b) Prove or disprove: the set fe, e2 is an orthonormal basis of V with respect to this nonstandard inner product. If it's not, perform the Gram-Schmidt orthogonalization process to find an orthonormal basis B for V. (c) Express an arbitrary vector x-(x1,x2) E V as a linear combination of basis vectors in an orthonormal basis that you found in (b). (d) Let W Span(-2,3)) c V. For an arbitrary vector x-(x1, r2) E V, determine the projection of x onto W.