Let u and v be vectors in a vector space V, and let H be any subspace of V that contains both u and v. Explain why H also contains Span {u, v}. This shows that Span {u, v} is the smallest subspace of V that contains both u and v.

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Using the orthogonal projection of vector Y to get the component of y along the vector space W.

Q: Find a linearly independent set of vectors that spans the same subspace of R' as that spanned by the…

A: Given :           Three vectors that spans of ℝ3 as :                  v1 = -23-2 , v2 = 480 , v3 =…

Q: Is the union of two subspaces of a vector space, a subspace.

A:

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: Given ,           W is a subspace spanned by u's      u1=1101   , u2=-131-2  , u3=-1011    &…

Q: Let (X, F) be a space. Prove that a subspace of a subspace is a subspace of (X, F).

A: Here given X, I is a topological space.

Q: If U is a proper subspace of a finite-dimensional vector space V,show that the dimension of U is…

A: We will prove this by taking help from two theorems. Please see the subsequent steps:

Q: c) Prove that if U and W are subspaces of a vector space V, then their intersection UnW is also a…

A:

Q: True or False: If U and W are subspaces of a finite dimensional vector space and UnW = {0}, then U°…

A:

Q: If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it

A: NOTE:  If V is a finite dimensional vector space and W is a subspace of V, then W is finite…

Q: Let H and G be subspaces of a vector space V. Show that HnG is a subspace of V.

A: We have to show that intersection of two subspaces is a subspace.

Q: Find a linearly independent set of vectors that spans the same subspace of R as that spanned by the…

A: A collection of vectors spans a set if every vector in the set can be expressed as a linear…

Q: Let W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A: In this question, the concept of the Orthogonal Decomposition theorem is applied. The vector…

Q: Find a linearly independent set of vectors that spans the same subspace of IR" as that spanned by…

A: The given vectors are                                   -3110, -12613, 3-55-6, -33-23…

Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U # W.…

A:

Q: show that if w is a subspace of a finite - dimensional vector space V and dim (w)= dim (V) then W =…

A: In this question, we will use the fact that If dim(W) = dim(V ) = n, then a basis for W is a…

Q: Find a linearly independent set of vectors that spans the same subspace of R* as that spanned by the…

A:

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors ui

A:

Q: Let U and W be subspaces of a vector space V. Show that U ∩ W isa subspace of V and that U + W = {u…

A:

Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

A:

Q: If V is direct sum of two subspaces, then every vector in V has .representation

A:

Q: Let V be a vector space and let V1, . . . , Vk be a collection of subspaces of V. Prove that V1 ∩ ·…

A:

Q: Let V be a vector space with subspaces U and W. Give example with v = R2 to show that U ⋃W need not…

A:

Q: Let V be a vector space and let u,v ∈ V. If W = {αu + βv|α,β ∈ R}, prove that W is a subspace of V

A: Let V be a vector space and let u,v∈V.W=αu+βv| α,β∈R

Q: 1. Let u and v be vectors in a vector space V, and let H be any subspace of V that contains both u…

A: As per guidelines we have to attempt only 1st question.

Q: The span of a set of vectors from a vector space V is a subspace of V. true or false?

A:

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors ui

A:

Q: Let V and W be subspaces of R² that are spanned by (3, 1) and (2, 1), respectively. Find a vector v…

A: From the given problem:   v+w=3,5 , V=3,1 & W=2,1   Let,    v=xVw=yW   So,…

Q: 16 In the vector space R, let u= (1, 2, 3), v = (3, 1, 5). w = (3,-4, 7). Prove that the subspace S…

A:

Q: Let ū be some fixed vector in R" and let S be the set of all vectors in R" that are orthogonal to ū.…

A:

Q: c) Find dimension of subspace of all vectors in R° whose first and third entries are equal.

A: We have to find the dimensions

Q: W is not a subspace of the vector space. Verify this by giving a specific example that violates the…

A: The objective is to prove that w is not a subspace of the given vector space.

Q: Let (V,(*,*)) be an F-inner product space,where F is either R or C. LetU C V be a subspace and let a…

A:

Q: Let W be the set of all the vectors of the form y where x and y are any real numbers. 2х-Зу…

A:

Q: (a) Determine whether the set of all vectors of the form (a,b,c), where b= a+c, is a subspace of R³.…

A: Consider the subset S of ℝ3 defined as, S=(a,b,c)∈ℝ3 | b=a+c Take  a1,a2,a3,b1,b2,b3…

Q: Suppose W is a subspace of Rn spanned by n nonzero orthogonal vectors. Explain why W = Rn.

A:

Q: ) Find a linearly independent set of vectors that spans the same subspace of IR" as that spanned by…

A: First we need to write given vectors in matrix form and try to use row reduced echelon form(RREF) :

Q: Is the union of two subspaces of a vector space, a subspace. Justify your answer.

A:

Q: he zero vector in U is a subspace of V.

A: Inroduction a. Zero vector is a subspace of every vector space.

Q: Prove that the intersection of two subspaces of a vector space is a subspace.

A: The objective is to prove that the intersection of two subspaces of a vector space is a subspace.…

Q: Find the orthogonal projection of u = 1 onto the subspace of R³ spanned by the vectors - 3 2 V1 =…

A:

Q: .Is the set of all vectors of the form b. where a and b are any real numbers, a subspace W of R?…

A: Yes, W is a subspace of R^3.

Q: Let V be the vector space of n-square matrices. Let U be the subspace of upper triangular matrices,…

A: Vector spaces

Q: Show that the set of vectors of the form a +b forms a subspace of R³. Find a 2а — b basis for this…

A: Given set of vectors of the form: aa+b2a-b Let v1=a1a1+b12a1-b1 and v2=a2a2+b22a2-b2 be two…

Q: In Exercise , find the orthogonal projection of v onto the subspace W spanned by the vectors u?.…

A: Given :           v = 31-2 ; u1 = 111 ; u2 = 1-10   To find an orthogonal projection of v onto the…

Q: Let ui = (1, –2, – 1), u2 = (2, 2, -2), and v = (0,0, 1). v is not in the subspace W spanned by u,…

A:

Q: Determine if the given Vector Subspace is LINEARLY INDEPENDENT IN THE REQUESTED VECTOR SPACE:

A:

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors ui

A: Given:   v = 123 , u1 = 2-21 , u2 = -114    Also, given that the subspace W is spanned by the…

Q: Consider the vector subspaces U, V  ,W ⊆ Rn such that U ⊆ V ⊆ W. If dimU = 5 and dimW = 7, what can…

A: - it always contains a set of 5  linear independent vectors (True) This statement is True. As dim V…

Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors u;. are orthogonal.)…

A: