1. Let V be a finite-dimensional inner product space and let wj, 10 vectors in V (with absolutely no other assumptions placed on these vectors). Define the subspace U as U = {v € V : (v,w;) = 0, 1< j< k}. (a) Show that U is a subspace. (b) Determine the dimension of U.

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

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: (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: Let U be an m-dimensional subspace of Rn andlet V be a k-dimensional subspace of U, where0 &lt; k…

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Q: Let W1 and W2 be subspaces of a vector space V having dimensions m and n, respectively, where m > n.…

A: Given, dimension of W1=mdimension of W2=n such that m≥n   We know, W1∩W2 is a subspace of W1W1∩W2 is…

Q: 2. Let U and W be subspaces of a vector space V such that UUW is a subspace of V. Prove that UCW or…

A: Given that U and W are subspaces of a vector space V such that union of U and W is a subspace of V.…

Q: 4. Decide if the following statement is True or False: Let U, W, and X be subspaces of a finite…

A: Given - Let U, W and X be subspace of a finite dimensional vector space V such that U ∩ W = 0 and U…

Q: #Consider the vector space R' (over R) and ZcR consisting of vectos z=(2,2, 2,), with these…

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

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

A: These 2 are unrelated problems since anyone can be solve without any help of other. By Bartleby…

Q: Consider the vector space V = {p(x) = ao +ajr+azx² | a¡ € R, deg(p(æ)) < 2} of polynomials of degree…

A: Let          V(x)=p(x)=a0+a1x+a2x2 | ai∈ℝ, deg(p(x))≤2 of polynomials of degree less than or equal…

Q: Let V be a vector space, F a collection of subspaces of V with the following property: If X,Y EF.…

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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 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: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

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

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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: 4. Use the definition of a subspace to prove that the set V:= :, y, z E R, y+ : 0 is a subspace of…

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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…

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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: Let V be finite dimensional vector space and let W c V be subspace with complimentary subspace W'.…

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Q: 4) Let X, Y, Z be subspaces of a vector space V and assume that Y CX. Prove that Xn(Y+Z)% =Y+(XnZ).…

A: 4. Given The modular law of sub-space: Let X,Y,Z be subspaces of a vector space V and assume that…

Q: Let W1 and W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that…

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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: Let V and W be vector spaces over R and let L:V → W be a linear mạp. Recall the kernel of L is…

A:

Q: 3. Prove that the set S is a subspace of the vector space V V = C(R) and S is the set of f in V such…

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Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

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

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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 V is finite-dimensional, with dim V = n > 1. Prove that there exist 1-dimensional subspaces…

A: Let us consider that (v1, v2, ..., vn) be a basis for V, and let Ui= span(vi), for i ∈ {1, 2, ...,…

Q: 4. Consider C" as a real vector space. It has a subspace U := C x {0}"-1. What is the dimension of…

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Q: W= im T when W is a T-invariant subspace, and V=ker + W. Where V is finite dimentional, when you let…

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Q: Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V,…

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Q: 2. Find a basis for the subspace W = {[r y z]T: ax + by + cz = 0, a, b, c e R} of R". What is the…

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Q: Prove that a subset W of a vector space V is a subspace of V if and only if 0 EW and ax+ y E W…

A: First proving the first part: If W is the subset of V and also the subspace of V Claim  : 0 W and…

Q: - Let V = R and W = {(a, b, c) E V \a + b = c}. Is W a subspace of V? If so, what is its dimension?

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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 V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V,…

A:

Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be 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: 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: 3) Show that the set of all elements of the form (x + 2y,x + 3z, 2x + y + z), where x, y, z E F, is…

A: We have, (x+2y , x+3z , 2x+y+z),where x, y, z € F.

Q: Let S be the set of all elements of the form (x + 2y , y ,  -x + 3y) in R3 , where x,y belongs to R…

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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: If B is basis for (x, t) then By = {BnY/BE basis for the subspace topology Ty on y IS 9

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Q: 4) Suppose u, v E V, where V is an inner product space, and ||u|| = |||| = 1 and (u, v) = 1. Find u…

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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: Let V and W be the subspaces of the vector space R^4 spanned by V1= (3,-1,4,1), V2 = (5,0,5,1), V3 =…

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Q: {{} } 5. Let W be the subspace of R³ spanned by write j= |3 as the sum of a vector in W and a vector…

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Q: 2. Let W be the subset of R' consisting of all vectors of the form (x, y, z) x= y+z. Prove that W is…

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Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.