Let W is a subspace of C and {u,..., u} be an orthonormal basis of W....

 

Transcribed Image Text:

Let W is a subspace of C and {u₁,..., u} be an orthonormal basis of W. Let v be an arbitrary vector in C. (a) (3 points) Show that (v - projwv) u; = 0 for all i and explain why v-projwv is in W¹. (b) (3 points) Show that every vector v in V can be written as v = W₁+W2₁ where w₁ is in W and w2 is in W. (c) (3 points) Prove that Wn W = ({0}. (d) (3 points) Prove that dim W + dim W¹ = n. (e) (3 points) If S is a subset of C", prove that (S) Spans Let W is a subspace of C and {u₁,..., u} be an orthonormal basis of W. Let v be an arbitrary vector in C. (a) (3 points) Show that (v - projwv) u; = 0 for all i and explain why v-projwv is in W¹. (b) (3 points) Show that every vector v in V can be written as v = W₁+W2₁ where w₁ is in W and w2 is in W. (c) (3 points) Prove that Wn W = ({0}. (d) (3 points) Prove that dim W + dim W¹ = n. (e) (3 points) If S is a subset of C", prove that (S) Spans

Answer rating: 100% (QA)

a To show that v projWv is in W where projWv is the projection of v onto W we need to prove that for every vector u in the orthonormal basis u u of W the inner product of v projWv with u is zero Lets ... View the full answer