1.Define basis for and dimension of a vector space V. Determine the dimensions of the subspace consisting of the vectors of the form (x, y, z, w) where w = y + z,x = y - z of R*.

Q: Let W be a subspace of the vector space V. Prove that the zero vector in V is also the zero vector…

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

Q: Show that V is a subspace of R³: V is the set of all (x, y, z) such that z = 2x + 3y

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Q: Prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and, whenever a ∈…

A: We need to prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and,…

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 and W be subspaces of a vector space V. 1. Prove that UnW is also a subspace of V. 2. Must 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: Given ,           W is a subspace spanned by u's      u1=1101   , u2=-131-2  , u3=-1011    &amp;…

Q: Let H be the set of points in the second quadrant in the plane V=R^2. That is, H = { (x,y) | x ≤ 0,…

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Q: Let S = {i, j} is a basis of R² and let W = {(x, y), x + y = 0} a- Show that W is subspace of R², b-…

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Q: Let U and W be the subspace of V, with V being the vector space on R body. UnW = {v : v € U ve v e…

A: handwriting solution

Q: Let T : V → W be a linear transformation from a vector space V into a vector space W. Prove that the…

A: As per our guidelines we are supposed to answer only first question. Please repost other question…

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: 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 W = {a + bx + cx? + dx³ € Pg : c = a – b+ 2d} be a subspace of P3. Then a basis for W is:

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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: and let W the subspace of R" spanned by i and v. Find a basis of W-, the orthogonal complement of W…

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Q: 5. Let U, W be subspaces of a vector space V such that (v, ) is a basis for U and (v2, es, 4) is a…

A: We shall use the formula dim(U+W)= dim U + dim W - dim (U \cap W) to solve this problem. The detail…

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: Let H be the set of all points in the second quadrant in the plane V=R^2. That is,…

A: Let H be the set of all points in the second quadrant in the plane V=R^2. That is,…

Q: Consider the vector space V = C2 with scalar multiplication over the real numbers R and let W and U…

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Q: Let H be a subspace of a vector space V, and suppose that V has dimension d. Which of the following…

A: a) Given a vector space V, let S=v1, ..., v2 be a basis for V. Consequently dimV=d.  Now given H, a…

Q: Suppose that V is finite dimensional and U is a subspace of V such that dim U = dim V . Prove that U…

A: Given that V is a finite dimensional vector space and U is subspace of V such that dimU=dimV. Let…

Q: Let W be a nonempty subset of a vector space V . Prove that W is a subspace of V if and only if r u…

A: Given: Let W be a nonempty subset of a vector space V

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: {[:} 5. (a) Consider the subspace W = span and consider the vector -1 1 u = -1 |. Find the vector w…

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Q: Let w = ( { (x.y,z) ER3: x+y-z 0) be a subspace. A basis of w+ is [A].

A: Solution

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

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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: Let V be a finite dimensional vector space and W CV a subspace. Prove that W has a complimentary…

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Q: Let H be the subspace of R with basis B = 3. and let x = 10 6. Is xa vector in H? 14 fx is in H,…

A: Given H is a subspace of R3 with basis {204,231}And x=10614To check x is in H

Q: Find a basis for the given subspace of R°, and state its dimension for the plane 3 x- 2 y+ 5 z = 0.…

A: We have the vector space R3 , the subspace here is: W={(x,y,z) | 3x-2y+5z=0 ; x, y, z∈R}

Q: Show that the nonempty subset W of a vector space V is a subspace of V if and only if for every pair…

A: Consider a vector space V over a field F. Suppose that W is a non empty subset of V such that…

Q: 1. Consider the vector space V = M3x3(C) over the field R. Let U be the subspace of V consisting of…

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Q: W = {(x, y, z): x 2 0},

A: Given the vector space

Q: Suppose the set {u,v} forms a basis for a subspace w in Rn. Let x=u+v and y=u-v. Show that {x,y} is…

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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 be a vector space and let u,v be elements in V. if W = { au + Bv | a,B are elements of real…

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Q: 3) Find a basis for the subspace of R that consists of the intersection of the planes X-3Y = 5Z and…

A: This is a problem of vector space.

Q: Let V=R^2 and let H be the subset of V of all points on the line 2x+3y=0. Is H a subspace of the…

A: We have given vector space V=ℝ2 and H is a subset of V of all points on the line 2x+3y=0. or we can…

Q: Let V=R^2 and let H be the subset of V of all points on the line 2x+3y=6. Is H a subspace of the…

A: Solution:Given :     V=ℝ2and      H= (x,y)/ 2x+3y=6   ⊂ ℝ2

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

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Q: Consider the vector space V = C° with scalar multiplication over the real numbers R and let W- (z1,…

A: Given is a space W in V. The objective is to prove that W is a subspace of V.

Q: True or False: If W is a subspace of a vector space V and B is a basis for V , then B can be…

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Q: Let V=R2 and let H be the subset of V of all points on the line 2x+3y=6. Is H a subspace of the…

A: Given : V=ℝ2 and H=x,y/2x+3y=6 We want to check is H a subspace of the vector space V

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: Prove that S = {(x, y, z) E R³: 2x + 3y+z = 0} is a subspace of the vector space R3.

A: Explanation of the answer is as follows

Q: (b) Consider the vector space V = {p(z) = ao + a1r+ azx² | a; € R, deg(p(x)) < 2} of polynomials of…

A: Consider the vector space V=px=a0+a1x+a2x2: ai∈ℝ, degpx≤2 and W=px∈V:p(1)=0

Q: Let V=R^3 and let H be the subset of V of all points on the plane -3x+5y-5z=0. Is H a subspace of…

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Q: i) Let V = M2x2(R) and %3D W; = {(: :) ev:a.hceR} and W2 = {(". :) ev :a,be R Prove that W1 and W2…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.