I EXAMPLE 7 Let V be a vector space over F and let v, V2, . .. , V, be (not necessarily distinct) elements of V. Then the subset (V, V2» ..., v,) = {a,v, + a,v, + · · · + a,v, \ q, a» - . . , a, E F} is called the subspace of V spanned by v,, v,, ..., v. Any sum of the form a,v, + a,v, + · · V1, V2, . .. , Vn. If (v1, v2, . spans V. n'n + a v, is called a linear combination of v) = V, we say that {v, V2, . .., V„}

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

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Q: Let U=span{uq ,U2,U3} Û=span {u, ,u,,u,} where ū,=(1,0,0,0), ū,=(0,1,1,0), ū,=(0,1,1,1) and…

A: (a) Given vectors are u1^→=1,0,0,1,u2^→=1,1,0,0,u3^→=0,0,1,1.   Consider U^=spanu1^→, u2^→, u3^→.…

Q: 1.Define basis for and dimension of a vector space V. Determine the dimensions of the subspace…

A: Spanning set of a vector space  Let us consider a set S=v1,v2,.....,vn then the spanning set S…

Q: linear algebra problem

A: According to the given information. Let P2 be the vectors space of degree 2 or less and H be the…

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: Show that [0,∞) and (0,∞) as subspaces of R with the usual topology are not homeomorphic.

A: Given: [0, ∞) and (0, ∞) are subspaces of ℝ. To show: [0, ∞) and (0, ∞)are not homeomorphic.

Q: 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…

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Q: 2. How can you build a subspace of a vector space V by using a single vector vEV

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Q: 4 Let S be the subspace of R* spanned by x1 = (1,0, – 2, 1)" and x2 Find a basis for S. (0, 1, 3,…

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Q: Let V be the subspace of R' spanned by the vectors below: u = (1,2, 1,3,2), Uz = (1,3, 3,5, 3), Uz =…

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Q: Let U and W be subspaces of R“ generated (spanned) by {{1, 1, 0, – 1], [1, 2, 3, 0), [2, 3, 3, –1]}…

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Q: 4. If V and W are sets, then their union VUW is the set consisting of all elements that are in V or…

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Q: 5.4) (6)  Find the projection of the vector v onto the subspace S. S = span 0 1 1 , 1 1 0 v = 3…

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Q: Theorem 6.8. Let A be a closed subspace of a compact space. Then A is compact.

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Q: if it was N-{cz,,z,»z,): 2z,-z, = o} Two subspaces of C³ space on field C find MON Then prove that…

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Q: The number of element of basis for a vector space V is called Select one: a. dimension of V b.…

A: Let V be a non-empty set and ⊕be a binary composition on V. Let F,+,. be a field and let ⊙ be an…

Q: What is the topology on the subspace {(x,1/2)|0 < or equal x < or equal 1} of the space [0,1] x…

A: The dictionary order on ℝ2 is defined as a×b&gt;x×y is a&gt;x or a=x and b&gt;y Here a×b and x×y are…

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…

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Q: Let the Euclidean space of 2x2 type matri with inner product defined by V, (A / B) = Isometry(B'A)…

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Q: Find the projection of the vector v onto the subspace S. 1 1 1 S = span 1 v = projs v =

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Q: A set of 4 vectors in R forms a subspace of R linearly dependent set linearly independent set basis…

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Q: 1. Find the ortho gonal puojection of the Vector (1,3,1 on the subspace of IR? Spanned by…

A: Projection

Q: Find an orthonormal basis for the subspace W of IR* spanned by th (а) vectors (1,4, ,…

A: (a) Given that (1,4,2),(-2,-1,3) and (6,-1,-1) This can be written as, A=142-2-136-1-1 So, let's…

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: 26 Let V be the vector space of all 2 x 2 matrices and let W be the set of matrices of the form X -x…

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Q: 39. Let 0 basis for W. be an orthogonal basis for a subspace W. Find an orthonormal

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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 &amp; 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…

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Q: Show that [0, 1] and (0, 1] as subspaces of R with the usual topology are not homeomorphic.

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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: Find the projection of the polynomial f(x) span{1, x} of C[0, 1] with the Sobolev inner product (f,…

A: The given function is fx=x3. The subspace V of C0, 1 is given as, V=span 1, x The Sobolev inner…

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…

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Q: 1. Find in three-dimensional vector space all subspaces invariant under the linear operator with the…

A: dear student we are only allow to solve first question if you want to rest of answere please upload…

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

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Q: 8. Let W denote the subspace of R consisting of all the vectors having coordinates that sum to zero.…

A: Consider the provided question, Write the given vectors into a matrix as below,

Q: Determine if the subset of C(- , 0) is a subspace of C(- ∞, 0). The set of all negative functions:…

A: We need to determine whether the subset of C(-∞,∞) is subspace of C(-∞,∞) or not. Here the function…

Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

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Q: 2. Let W be the subspace of R5 spanned by the vectors (1,1,1,1,1),(1,0,1,0, 1), (0,1,1,1,0),…

A: For the solution of the problem follow the next steps.

Q: Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.

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Q: 4. Find a basis for the intersection of the subspaces V = Span ((1, 0, 1, 1), (2, 1, 1, 2)) and W =…

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Q: (1, 2, 3, 4, 6) and let W be the subspace of R° spanned by {u1, u2}, where u1 (1, 2, 1, 2, 1) and uz…

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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: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

A: Given that H is a subspace of the vector space P2 of all polynomials of degree 2 or less spanned by…

Q: Construct an orthonormal basis for the subspace of R spanned by the vectors (1, 2, 3, -1), (2, 0,…

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Q: TA Let X- (1,10,14) and - 0,x,(1), (10). [1,10}} be a topology on x. Given that 4= (10,14, compute…

A: Let X=1, 10, 14 and T=φ, X, {1}, {10}, {1,10} the topology on X. Given A=10,14 Subspace topology of…

Q: If V is vector space that has a basis contain n vectors, then every subspace of V must only have…

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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,…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.