11. Consider the linear space R with its standard scalar product (,) R R R defined...

 

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11. Consider the linear space R³ with its standard scalar product (,) R³ R³ R defined as ((x, y, z), (x, y, z)) : = xx' + yy' + zz. Let a, b, c be the last three non-zero digits of your (7-8 digit) student number. Consider the ordered basis ß = {u1, U2, U3} for R³, where U₁ = 0,4₂ = 3 , Uz = To] 0 C Fill in the values a, b, c to give your unique ordered basis 3. (a) Determine the orthogonal complement U of U = span({u₁, ₂}). (b) Apply the Gram-Schmidt procedure to 3 to construct an orthonormal basis y {V₁, V2, V3}. (c) Let w R³ be a vector with coordinates given with respect to the standard basis for R3. Prove that the coordinates of w with respect to an orthonormal basis y = {V₁, V2, V3) are (w. v₁, w v2, W. v3). [6] [14] [5] 11. Consider the linear space R³ with its standard scalar product (,) R³ R³ R defined as ((x, y, z), (x, y, z)) : = xx' + yy' + zz. Let a, b, c be the last three non-zero digits of your (7-8 digit) student number. Consider the ordered basis ß = {u1, U2, U3} for R³, where U₁ = 0,4₂ = 3 , Uz = To] 0 C Fill in the values a, b, c to give your unique ordered basis 3. (a) Determine the orthogonal complement U of U = span({u₁, ₂}). (b) Apply the Gram-Schmidt procedure to 3 to construct an orthonormal basis y {V₁, V2, V3}. (c) Let w R³ be a vector with coordinates given with respect to the standard basis for R3. Prove that the coordinates of w with respect to an orthonormal basis y = {V₁, V2, V3) are (w. v₁, w v2, W. v3). [6] [14] [5]

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