6. Let V be a vector space and let U1 and U2 be subspaces of V. Show that if U1 U U2 is a subspace of V then either U1 C U2 or U2 C U1. Hint: Show that if U1 is not contained in U2 and U2 is not contained in U1 then U1 U U2 is not closed under addition.

Q: (b) If V = R", and U is a subspace of V of dimension m with 0 <m <n, prove that U² = {v € V: B(u, v)…

A: As per our guidelines we are supposed to solve only one problem. For the solution of the first…

Q: Prove that if W is a subspace of a vector space V and Wt, w2, ... , wn are  in W, then a1Wl +a2W2 +…

A: To Proves: If W is a subspace of a vector space V and w1,w2,...,wn are in W, then…

Q: In each item if W is a real vector subspace of V: a) V= M2 (IR) and W = {A € V |A is not…

A:

Q: Suppose U1,U;, -..Um are finite-dimensional subspaces of V. Prove that U + Uz + .. +Um is…

A: Suppose U1, U2, ... , Um are finite-dimensional subspaces of V. To prove U1+U2+ ... +Um is finite…

Q: Consider the following subspaces of R3: U=span[(2, 1, 1), (1, 2, 0)] W=span[(0, 2, 1), (0, 1, 2)]…

A:

Q: If u and v are vectors in R and W is a subspace of R with (u - v) belong to W, then both u and v…

A: If u and v are vectors in R3 and W is a subspace of R3 with (u-v) belong to W, then 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: Given W is a subspace spanned by the u's where y=322-1, u1=1101, u2=-120-1 AND u3=-1011   To Write:…

Q: Prove that if W is a subspace of a vector space V and w1, w2, . . . , wn are in W, then a1w1 + a2w2…

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: We need to find y.

Q: Prove that the set of vectors of the form (a, b, c) where b = a +c is a subspace of R³.

A: According to the given information, it is required to prove that:

Q: . Let u₁ = (1, -2, -1), U₂ = (2, 2,-2), and v = = (0, 0, 1). v is not in the subspace W spanned by…

A: In this question, a vector that is orthogonal to a given vector is to be determined. A non-zero…

Q: The set S = ( (x.y, 2) ER |2 = 2, x,y,ZER)is a subspace under vector addition and scalar…

A: This question is related to linear algebra topic vector space.

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: [5 Find the orthogonal projection ŷ of y = onto the subspace 3 W = Span { uį = 2 u2 3 Ex: 1.23 ŷ =

A:

Q: Let-{ », :x, y, a, b e R and x - a = y – b}.Is W a subspace?

A: We will use the one step test for vector subspace to check if W is a subspace or not. We know that…

Q: Let X and Y be a topological spaces and assume that A C X and B CY. Then the topology on Ax B as a…

A:

Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

A:

Q: Which of the following choices is not a subspace of vector space R? a. (x,y,z):z=x-y b.…

A:

Q: Let u, = 2 |, u2 and uz = 0 Note that u, and uz are orthogonal but that uz is not orthogonal to u,…

A: The orthogonal vector is given as follows :

Q: Show that the set S = {(x, y) in R2 such that x =3y } is a subspace of the vector space R2 NOTE!…

A: That's easy. Have a good day!!!

Q: Suppose U and W are two-dimensional subspaces of R3. Show that U∩W≠{0}

A: We want to show that  U∩W≠{0}

Q: Consider the following subspaces of R3: U=span[(2, 1, 1), (1, 2, 0)] W=span[(0, 2, 1), (0, 1, 2)]…

A:

Q: Let V be a vector space and U1,u2,U3,Ug are vectors in V. What is the dimension of the subspace S=…

A:

Q: Let W fa + bx + cx2 + dx'| a+ b 0 and c-3d 0 } be a subspace of Pg. %3D %3D %3D Then the dimension…

A: a+b=0  and c-3d=0 b=-a and c=3d Equation given is; a+ bx+ cx2+ dx3  = a- ax+ 3dx2+ dx3 that can be…

Q: Let W is a finite dimensional subspace of an inner product space V and y is any vector in V. The…

A: Let W be a finite dimensional subspace of an inner product space V. Let T be an orthogonal…

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: Consider the vectors v1, v2, ..., Um in R". Is span (01,..., īm) necessarily a subspace of R" ?…

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:

Q: 06. Suppose V is an inner product space, and let U,, U, be two subspaces of V satisfying dim (U,) <…

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:

Q: Provide a situation in where it shows a union of two subspaces of a vector space V and is not…

A:

Q: If W is a subspace of R" and if u is in both W and w then u must be a zero vector.

A: I am going to solve the given problem by using the orthogonal property of a vector.

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:

Q: Suppose that A is a fixed 2x2 matrix. Let U = {X ∈ M22 | AX = XA} Question: Prove that U is a…

A:

Q: 13. Let U, W be vector subspaces of a vector space V. If dim(U) 3, dim(W) 4, dim(V) = 7 and UnW =…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Consider X = {1, 2, 3, 4, 5} where the topology is {U C X|1 E U}. Find a subspace of X which is…

A: Consider the set X = {1,2,3,4,5} where the topology is {U ≤ X : 1 ∈U}. We have to find a subspace of…

Q: ) Do there exist subspaces S,T, and U of R³ such that S 1 T, SI U, and TiU? if so, provide an…

A: I have considered te subspace generated by orthogonal basis vector.

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

Q: Let u, = u2 = -1. and uz =0| Note that u, and uz are orthogonal but that uz is not orthogonal to…

A: Introduction: If two vectors are perpendicular to one other, they are said to be orthogonal. The dot…

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 vectors are y=466−1, u1=1101, u2=−120−1 and u3=−1011. We have to write the vector y as a sum…

Q: Let U1, U2, U3 - be the following subspaces of R* Ui={(a, b, c, d)eR*| a - b + .2c + 3d=0, a - Ob -…

A:

Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: To prove: If V and W are subspaces of ℝn such that dim V &lt; n, dim W &lt; n, V is not contained in…

Q: If y is not in a subspace W, then y - ŷ is zeo. Select one: True False

A: If y is in subspace W, then the orthogonal projection of y onto W (y') is y itself.  And in that…

Q: 2. Let U and W be a subspaces of an n-dimensional vector space V and suppose further that UNW={0}…

A:

Q: 2. Let S = {vi, v2, V3} be a set of linearly dependent vectors in an inner product space V. What is…

A: To explain: What will be the result when we apply Gram-Schmidt process to a linearly dependent set…

Q: Prove that the intersection W n U of two subspaces of a vector space V is a subspace of V and prove…

A:

Q: Let W be the subspace of R5 orthogonal to u1=(1, 3, -2, 1, 1) and u2=(0, 1, 1, -1, 0), u3=(0, 0, -2,…

A: 1.Let W be the subspace of R5 which is orthogonal to S=1,3,-2,1,1,0,1,1,-1,0,(0,0,-2,1,2)…

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: Prove that is U1 and U2 are subspaces of V then dim(U1 + U2) dim(U1)+dim(U2)-dim(U1 N U2). Using…

A: