6a. Suppose that B is a basis of V. Decompose it as a disjoint union B = B₁ I IIB₂ of non-empty sets B₁. Show that B, is a basis of W; := Span(B₂) and V=W₁ 0... Ws is a direct sum of nonzero vector subspaces W₁ of V. 6b. Conversely, suppose that V = W₁ W, is a direct sum of nonzero vector subspaces W₁ of V. Let Bį be a basis of W;. Show that B = B₁ III B₂ is a basis of V and a disjoint union of non-empty sets B₁.

Q: 24. Let S be a basis for an n-dimensional vector space V. Show that if v, v2. ...,v, form a linearly…

A:

Q: 15. Let V be the set of those polynomials ax² + bx + c € P₂ such that P2 a + b + c = 0. Is V a…

A: Let V be the set of those polynomials ax2+bx+c∈P2 such that a+b+c=0. To determine whether V is a…

Q: V is a 3-dimensional vector space with basis By (V1, V2, V3) and W is a 3-dimensional vector space…

A:

Q: 2-Show that the following polynomials from a basis for P2. 1- 3x + 2x²,1 + x + 4x²,1 – 7x

A: Consider the polynomials: 1-3x+x2, 1+x+4x2, 1-7x To show that the following polynomials form a basis…

Q: 1. Consider the following vectors in R4: 2 1 H 0 The set B = {V₁, V2, V3} is a basis of some…

A: We are given set is a basis of some subspace V of where To find orthogonal basis of V, lets use…

Q: Which of the following sets of vectors is a basis for the subspace W of R' given by W = ¿x = z + :…

A:

Q: Consider the subspace S of the Euclidean inner product space R* spanned by the vectors v₁=(1,1,1,1),…

A:

Q: 3.) Consider the following vector. O M 2 2 t 2 6 x3 = 16 -S a) Are the vectors {x₁, X₁, X3) lineonly…

A: Since you have posted multiple sub parts. according to guidelines, I solve only first three…

Q: 3- Let {u,v,w} be a basis for a vector space V. Is {u + v,u + w,v + w} a basis for V?

A: Solution

Q: Let W₁ = G0---60) and W₂ = (a) Find a basis for W₁ W₂. What is dim(W₁ W₂)? (b) Show that R³ = W₁ +…

A:

Q: If W is a subspace of R" of dimension k with a basis {v1,e,2,..., v%} then W Range(|v1 v2 vn)). ...…

A: Given: Let W be a subspace of Rn and dimk=k Let B be a basis of k then B=k Also B is a linearly…

Q: ra Which of the following is a basis for the subspace U = O -08 a) -88 d) e) HB 88 c) - 2b = c } <R²…

A:

Q: 1. Let W = {( 2).a+ 2d = 0}. Show that W is a subspace of the set of all 2 x 2 matrices and find its…

A:

Q: Find a basis of the subspace of Rª that consists of all vectors perpendicular to both 1 1 and -6 2 마…

A:

Q: a) Give an example of subspaces U = span(f1, f2, f3) and V = span(g1,92; g3) of RaJa] such that…

A:

Q: Let {u,., up} be an orthogonal basis for a subspace W of R" and {vr, ., vg} be an orthogonal basis…

A:

Q: Let -B 5 Find a basis of the subspace of consisting of a vectors perpendicular to U=

A: Orthogonal Complement: Let A be a matrix and let W=Col(A). Then the orthogonal complement of W is…

Q: You need to provide a clear and detailed solution for the following Questions: Question Let W be the…

A:

Q: Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that v1, V2, V3…

A:

Q: 1. Find a basis for the subspace of R4 spanned by the vectors (1,1,-4,-3), (2,0,2,-2), (2, -2,3,2)

A: Solution:-

Q: a) Supppose H is a 2 dimensional subspace of R³ with a basis {b₁,b2}, and y is a vector given by b₁…

A:

Q: Find a basis for the subspace of R° spanned by the following vectors. 6. 18 -3 -3 6 6. -18 -18 14…

A: We solve this by row reduced echolon  form

Q: 1-Determine if W is a subspace of Mnn CEMnn, W = {AEMnn : CA'C²= A} %3D

A: It is given that W=A∈Mnn : CAtC-1=A, where At is the transpose of the matrix A. We have To determine…

Q: Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that V1, V2, V3…

A: We shall answer first question only as you’ve asked more than one question. For others kindly post…

Q: Show that the vector space P, has the basis S = {1, x, x², x³}. %D

A: We have to solve given problem:

Q: A finite set of vectors v,,...,v, is called a linear basis if a. This set generates vector space…

A: Basis of the vector space

Q: 11. Let V be an inner product space and let B1, ..., B, be any independent vectors in V. The may…

A:

Q: Consider a subspace and a vector (:) (3) 1 H = Span- 2 X = %3D 2 7 Determine if x e H. If so find…

A:

Q: Show that if U and W are subspaces of the vector space V, then dim UnW < min{dim U, dim W} . Prove…

A: Given statement is false. Counter example is as follows

Q: 3) Suppose that U and W are subspaces of a vector space V with V = U €W. Suppose also that u, ., Um…

A:

Q: 1-Determine if W is a subspace of Mnn CEMn, W = {A€Mn: CA°C1 = A}

A: W is a subspace of V if:  u,v∈W ⇒ u+v∈W au∈W, a  is a scalar Now, we have W=A∈Mnn : CAtC-1=A Let…