x. y - Fix n E N. Given x, y E R, recall that the dot...

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x. y - Fix n E N. Given x, y E R", recall that the dot product x y ER is defined by ₁. Prove the following for any x, y, z = R" and a € R: 1 (a) x. x = |x||² (b) x· y =y.x (c) x (y +z) = x.y +x.z (d) (ax) y = a(x · y) = x. (ay) (e) ||ax|| = |a|· ||x|| (f) ||x ±y||² = ||x||² ± 2x · y + ||y||² (g) If x = ||x||−¹ . x, then ||Â|| 1. = (h) x y ≤ ||x||· ||y|| (“Cauchy-Schwarz in- . equality") (i) x +y|| ≤ Hint: For (h), first show that the statement is equivalent to x ||x ±ý||² ≥ 0. For (i), use that squaring both sides yields an equivalent statement. x|| + ||y|| ≤ ||*||· ||ŷ||. Then use that x. y - Fix n E N. Given x, y E R", recall that the dot product x y ER is defined by ₁. Prove the following for any x, y, z = R" and a € R: 1 (a) x. x = |x||² (b) x· y =y.x (c) x (y +z) = x.y +x.z (d) (ax) y = a(x · y) = x. (ay) (e) ||ax|| = |a|· ||x|| (f) ||x ±y||² = ||x||² ± 2x · y + ||y||² (g) If x = ||x||−¹ . x, then ||Â|| 1. = (h) x y ≤ ||x||· ||y|| (“Cauchy-Schwarz in- . equality") (i) x +y|| ≤ Hint: For (h), first show that the statement is equivalent to x ||x ±ý||² ≥ 0. For (i), use that squaring both sides yields an equivalent statement. x|| + ||y|| ≤ ||*||· ||ŷ||. Then use that