Problem 2. (40 points) Let there be the vector space (R, R) and subspaces V, W...

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Problem 2. (40 points) Let there be the vector space (R³, R) and subspaces V, W CR³ V = W = X |-x+y+z=0, x, y, z ER R |-x=y=z, x, y, z ER (a) Prove that for any vector v EV, w W, (v, w) = vw = 0. (b) What are the dimensions of V and W? (c) Compute the sum Z of subspaces V and W, i.c. Z = V + W. (d) Prove that VnW=0g. Does this imply that VW = R³? Please explain why. (e) Let there be the following optimization problem min ||r-w||₂, x which can be expressed as the following optimization problem a BEL min WEW Of the sets V and W, where does z → min 3a² + 2a + 1 aЄR belong to? Please explain why. Problem 2. (40 points) Let there be the vector space (R³, R) and subspaces V, W CR³ V = W = X |-x+y+z=0, x, y, z ER R |-x=y=z, x, y, z ER (a) Prove that for any vector v EV, w W, (v, w) = vw = 0. (b) What are the dimensions of V and W? (c) Compute the sum Z of subspaces V and W, i.c. Z = V + W. (d) Prove that VnW=0g. Does this imply that VW = R³? Please explain why. (e) Let there be the following optimization problem min ||r-w||₂, x which can be expressed as the following optimization problem a BEL min WEW Of the sets V and W, where does z → min 3a² + 2a + 1 aЄR belong to? Please explain why.

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a For any vector v EV w W v w v ... View the full answer