Let w,v1,...,vp ∈ Rn and suppose that w ∈ Span{v1,...,vp}. Show that Span{w,v1,...,vp} = Span{v1,...,vp}. Let...

Let w,v1,...,vp ∈ Rn and suppose that w ∈ Span{v1,...,vp}. Show that Span{w,v1,...,vp} = Span{v1,...,vp}.
Let v1,...,vp,w1,...,wq ∈ Rm. Is the following statement True or False?

“If {v1, . . . , vp} is linearly dependent then {v1,...,vp,w1,...,wq}

is linearly dependent.”
If you answer True, provide a complete proof; if you answer False, provide a counter-example. Linear Algebra. Please show both!

Expert Solution

Colby Messinger answered 7 months ago

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

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 W be an inner product space and v1, . . . , vn a basis...

Let W be an inner product space and v1, . . . , vn a basis of V . Show that <S, T> = <Sv1, T v1> + . . . + <Svn, T vn> for S, T ∈ L(V, W) is an inner product on L(V, W).

Let T,S : V → W be two linear transformations, and suppose B1 = {v1,...,vn} andB2...

Let T,S : V → W be two linear transformations, and suppose B1 = {v1,...,vn} andB2 = {w1,...,wm} are bases of V and W, respectively. (c) Show that the vector spaces L(V,W) and Matm×n(F) are isomorphic. (Hint: the function MB1,B2 : L(V,W) → Matm×n(F) is linear by (a) and (b). Show that it is a bijection. A linear transformation is uniquely specified by its action on a basis.) need clearly proof

Let T : Rn →Rm be a linear transformation. (a) If {v1,v2,...,vk} is a linearly dependent...

Let T : Rn →Rm be a linear transformation. (a) If {v1,v2,...,vk} is a linearly dependent subset of Rn, prove that {T(v1),T(v2),...,T(vk)} is a linearly dependent subset of Rm. (b) Suppose the kernel of T is {0}. (Recall that the kernel of a linear transformation T : Rn → Rm is the set of all x ∈ Rn such that T(x) = 0). If {w1,w2,...,wp} is a linearly independent subset of Rn, then show that {T(w1),T(w2),...,T(wp)} is a linearly independent...

Let B be a basis of Rn, and suppose that Mv=λv for every v∈B. a) Show...

Let B be a basis of Rn, and suppose that Mv=λv for every v∈B. a) Show that every vector in Rn is an eigenvector for M. b) Hence show that M is a diagonal matrix with respect to any other basis C for Rn.

Let S be a subset of a vector space V . Show that span(S) = span(span(S))....

Let S be a subset of a vector space V . Show that span(S) = span(span(S)). Show that span(S) is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all subspaces of V that contain S as a subset.

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 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 A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC)...

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC) = rank(C),

Question