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Instructions. Use the attached MS Form as your answer sheet. Do not upload this exam on any website. 1. Let {w1​,w2​,w3​} be an orthonormal basis for V. Given that v−α1​v1​+α2​w2​+α3​α3​v2​+α2​v3​. If the angle between v and w1​ is θ−60∘ and ∥v∥−A, what is the value of α1​ ? (a) 1 (b) 2 (c) 3 (d) 1 2. Equip R2​[x∣ with the inner product ⟨p(x)∣q(x)⟩−∫01​p(x)q(x)dx. If projx​x2−αx for some α∈R, what is the value of α ? (a) 1/4 (b) 1/2 (c) 3/4 (d) 1/8 3. The Gram-Schmidt process was applied to v1​−⎝⎛​111​⎠⎞​,v2​−⎝⎛​110​⎠⎞​ and v3​−⎝⎛​100​⎠⎞​ obtaining the vectors w1​,w2​ and w3​. Which of the following is equal to w3​ ? ⎝⎛​2/30−2/3​⎠⎞​ (b) ⎝⎛​2/3−2/30​⎠⎞​ (c)⎝⎛​1/20−1/2​⎠⎞​ (d) ⎝⎛​1/2−1/20​⎠⎞​