Let W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.

Let W be a subspace of Rⁿ. Prove that W⊥ is also a subspace of Rⁿ.

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Colby Messinger answered 2 years ago

Let W be a subspace of Rn with an orthogonal basis {w1, w2, ..., wp} and...

Let W be a subspace of Rn with an orthogonal basis {w1, w2, ..., wp} and let {v1,v2,...,vq} be an orthogonal basis for W⊥. Let S = {w1, w2, ..., wp, v1, v2, ..., vq}. (a) Explain why S is an orthogonal set. (b) Explain why S spans Rn. (c) Showthatdim(W)+dim(W⊥)=n.

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker...

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker P is W^perp.

Let U be a subspace of V . Prove that dim U ⊥ = dim V...

Let U be a subspace of V . Prove that dim U ⊥ = dim V −dim U.

A subspace of Rn is any set H in Rn that has three properties: a) The...

A subspace of Rn is any set H in Rn that has three properties: a) The zero vector is in H. b) For each u and v in H, the sum u + v is in H. c) For each u in H and each scalar c, the vector cu is in H. Explain which property is not valid in one of the following regions (use a specific counterexample in your response): a) Octant I b) Octant I and IV...

Let V -Φ -> W be linear. Show that ker (Φ) is a subspace of V...

Let V -Φ -> W be linear. Show that ker (Φ) is a subspace of V and Φ (V) is a subspace of W.

Let (V, ||·||) be a normed space, and W a dNormV,||·|| -closed vector subspace of V....

Let (V, ||·||) be a normed space, and W a dNormV,||·|| -closed vector subspace of V. (a) Prove that a function |||·||| : V /W → R≥0 can be consistently defined by ∀v ∈ V : |||v + W||| df= inf({||v + w|| : R≥0 | w ∈ W}). (b) Prove that |||·||| is a norm on V /W. (c) Prove that if (V, ||·||) is a Banach space, then so is (V /W, |||·|||)

2) Let v, w, and x be vectors in Rn. a) If v is the zero...

2) Let v, w, and x be vectors in Rn. a) If v is the zero vector, what geometric object represents all linear combinations of v? b) Same question as a), except now for a nonzero v. c) Same question as a) except now for nonzero vectors v and w (be care- ful!). d) Same question as a) except now for nonzero vectors v, w, and x (be extra careful!).

Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are vectors in W. Suppose...

Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are vectors in W. Suppose that v1; v2 are linearly independent and that w1;w2;w3 span W. (a) If dimW = 3 prove that there is a vector in W that is not equal to a linear combination of v1 and v2. (b) If w3 is a linear combination of w1 and w2 prove that w1 and w2 span W. (c) If w3 is a linear combination of w1 and...

Let x ∈ Rn be any nonzero vector. Let W ⊂ Rnxn consist of all matrices...

Let x ∈ Rn be any nonzero vector. Let W ⊂ Rnxn consist of all matrices A such that Ax = 0. Show that W is a subspace and find its dimension.

Prove the following: Let V and W be vector spaces of equal (finite) dimension, and let...

Prove the following: Let V and W be vector spaces of equal (finite) dimension, and let T: V → W be linear. Then the following are equivalent. (a) T is one-to-one. (b) T is onto. (c) Rank(T) = dim(V).

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