2 \u03c8 e 3 e 3 e 1 0 e 2 5 2 e 3 1 5 2 6 x 2 6 x 1 15 x 2 15 x 3 2 8 ii Anything

2 ψ e 3 e 3 e 1 0 e 2 5 2 e 3 1 5 2 6 x 2 6 x 1 15 x

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2 + ψ ( e 3 ) e 3 = e 1 + 0 e 2 + ( - 5 / 2) e 3 = 1 - 5 2 (6 x 2 - 6 x + 1) = - 15 x 2 + 15 x - 3 2 . 8. (ii) Anything orthogonal to u U and to w W is orthogonal to their sum u + w , so U W ( U + W ) . On the other hand, since U U + W and W U + W , it follows that ( U + W ) U and ( U + W ) W . Therefore, ( U + W ) U W . (i) This follows from (ii), replacing U and W by U and W : U + W = ( U + W ) = ( U ) ( W ) = ( U W ) .
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