2) Let W1, W2 be subspaces of V. Prove that W1 UW2 is a subspace if and only if W1 C W2 or W2 C W1.

Q: (2) Find a basis for the subspace W = {(a +c, 2a – b, 3a – 26 – c,b+ 2c) | a, b, c € R} of R.

A: We need to find the basis for the given subspace:

Q: Let V = M2x2. W. (R) = € V : a, b, cER W2(R) € V : a € R then prove that W, and W2 are subspaces of…

A: Necessary condition to be a subspace is: Vector addition property Scalar multiplication property.…

Q: If V = W1 ⊕ W2 and W is any subspace of V such that W1 ⊆ W , show that W = W ∩ W1 ⊕ W ∩ W2.

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Q: Determine if the set W of vectors (x1 , x2) in R such that xr? + x2 = 1 is a subspace of R’.

A: W is not a subspace of R²

Q: Suppose V is an interior multiplication space with finite dimension and W1 and W2 are subspaces of…

A: suppose that V is an interior multiplication space with finite dimension and W1 and W2 are subspaces…

Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

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Q: If S is a subspace of R", then so is the orthogonal complement of S.

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Q: Determine: If W = the set of all vectors in R2 whose second component is the square of the first. is…

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Q: Let W be the subspace of P3 spanned by {t³ +t² – 2t + 1, t² + 1, t³ – 2t, 2t³ + 3t² – 4t + 3}. Find…

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Q: Let W {(a,b, c, d)| 5a - c = 0 } be a subspace of R*. Then a basis for W is

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Q: Let vị 2,v2 8,and w Determine if w is in the subspace of R generated by v1 and v2.

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Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

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Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

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: 1. Show that WW10 W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Let W { (а, b, а + 2b + с, 2а — Зc), а, ь, с€ R} %3D - 1. Show that W is a subspace of R4. 2.…

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Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

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Q: Let W {(a, b, c, d)| a – 5 b = 0 } be a subspace of R*. Then a basis for W is

A: The given subspace of R4 is W={ ( a , b  ,c , d ) | a - 5 b = 0 } Let x=( a, b , c , d ) be in W…

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: 16. Show that the set V = {(a-2b+c, 3b, c +2a, b-c) | a, b, c = R} is a vector space that is a…

A: as you asked to solve question 16 only:   given: V = {(a-2b+c, 3b, c+2a, b-c) | a, b, c are in R}…

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: If W is a subspace, then ||projwv||² + ||v – projwv||2 = ||v|?.

A: Given W is a subspace of a vector space V. Now projWv denotes the projection of vector v onto the…

Q: Show that W = {a + bx² € P₂|a, b = R} is a subspace of P2.

A: The solution is given as

Q: Prove that if U and V are subspaces of R" and U <V then V+<U+.

A:

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: Show that if W₁ and W₂ are subspaces, then W₁ U W₂ is not a subspace unless one is a subspace of the…

A: We have to prove that if W1 and W2 are subspaces, then W1∪W2 is not a subspace unless one is a…

Q: 2. Provide an example of a subset U of V that is a subspace of V, as well as an example of subset W…

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Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that ,  spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

Q: If V is a vector space and W is a subset of V that contains the zero vector then W must be a…

A: Given statement If V is a vector space and W is a subset of V that contains the Zero vector and W…

Q: i) Determine whether the set W = {(x1, x2, x1x2) : xị and r2 are real numbers} is a subspace of R³…

A: Introduction: We prove the given set to be valid as per the standard conditions:

Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

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Q: (b) Define subspace and Verify whether is subspace v = {(x,y,2):x,y,2ER² and x-2y+3z = 5}

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Q: Suppose that X = eR' : 2 + y + z = 0 and r²+y² = z² } . Determine (with justification) whether X is…

A: We have to check whether the set X is a subspace of the vector space ℝ3. We have to give a…

Q: 2. Determine which of the following is a subspace of R". (a) {(x1,x2,...X) 1 x = 0} (b) {(x1,…

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

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Q: Show that S = {(a,0, b)"|a,b E R} forms a subspace of R³.

A: The solution are next step

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 V = M22 and W = {invertible 2 x 2 matrices}. Determine whether W is a subspace of M2. %3D

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Q: 3. Consider V = {a2x² +a1x + ao € P2(R) : a2 + a1 + ao = 0} . Use the subspace theorem to show that…

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Q: let W = {x €R3 , 1 such that Ax 2 where A is any 2×3 matrix is a subspace of R³ Select one: True…

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Q: If V = M2x2(R) and W consists of those elements of V in which at least one entry of the matrix is…

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

Q: Determine whether the set W is a subspace of R3 with the standard operations. w = {(x1, 0, x3): X1…

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Q: Determine: if W = the set of all vectors in R2 whose components are integers is a subspace.

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Q: Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1.…

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Q: Determine: if W = the set of all vectors in R^2 whose second component is the square of the first.…

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Q: Let W1,W2,K1,K2, . . . ,Kp,M1,M2, . . . ,Mq be subspaces of a vector space V such that W1 = K1⊕K2⊕·…

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Q: 2. Determine whether W is a subspace of V = C³ where W = {(0, b, c) E C3}

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

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