Let v1, V2, . , Vn E R". Then S = span(v1, V2, ., Vn) is a subspace in R". show that 1.0 is in the space, 2. it is closed under addition 3. closed under scalar multiplication).

Q: Prove thatII = {(x, y, z)|x – 2y+ 3z = 0}is a subspace of JR³, %3D -

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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: Let S = { (x, y, z, w) E R*; x – 2y + w = 0 and 2x + z – w = 0 } . - Prove that S is a subspace by…

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Q: Let W be the subspace of R3 spanned by the following five vectors: u1=(0, 1, 0), u2=(1, 2, 0),…

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Q: Let U be an m-dimensional subspace of Rn andlet V be a k-dimensional subspace of U, where0 < k…

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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    &…

Q: b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U…

A: In a finite-dimensional Vector space, every subspace is finite-dimensional.

Q: Let V be the subspace of the vector space of all real-valued functions that is spanned by the setS…

A: If V is the subspace of the vector space of all real-valued functions that is spanned by the set S…

Q: HConsider the vector space R° (over R) and ZcR consisting of vectors z=(z1, z2 z3), with these…

A: Given Z⊂R3, where R3 vector space.z→=(z1,z2,z3)(a) z1=0(b) z1+z2=0(c) z1+z2=1We have to check in…

Q: Let V = {(x, y, 2x – 3y)|x, yɛR}. Prove that V is closed for addition and scalar multiplication i.e.…

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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: Suppose W,, W2,. (a) span (W, W2 (b) If S; spans W; for i = 1,...,r, then S, US, U...US, spans W, +…

A: (a) W1+W2W1+W2 would be W2W2. Below is the justification.  x+y∈W1+W2,wx+y∈W1+W2where x∈W1x∈W1 and…

Q: Let V be a vector space and let u, v E V. If W = {au + Bula, BE R}, prove that W is a subspace of V.

A: Given :W be the non empty subset of V satisfying conditions α,β∈R and αu+βv∈W

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

A: Given:- V and W are subspaces of ℝn              dim V&lt;n, dim W&lt;n              Also, there is…

Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Let V be a vector space over a field F, and let W be a subset of V. Prove that Wis a subspace if and…

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Q: Is the subset V {(x, y, z, w), y = 3z, w = 2x + 1} a subspace of R*? Show the details of %3D your…

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Q: Let U and W denote subspaces of Rn, and assume that U ⊆ W. If dim U = n − 1, show that either W = Rn…

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Q: Let S consists of all vectors of the form (x, 0, y, 0) in R* whe y are real numbers. Determine…

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

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Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

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Q: Let W be a subspace spanned by the u's, an

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

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

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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: Let V be a complex vector space such that dim V = n and let T E L(V) be diagonalizable. We define…

A: Given U={S∈LV:ST =TS} Let V ={v1,v2,…,vn} be a basis for V consist of eigen vectors. Let λj is the…

Q: is

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Q: If S, is a subspace of R* of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠ S2 and S2≠R4.

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: If S is a set of vectors in an inner product space V(F) then prove that S- is a subspace of V.

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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: Show that the set W={(x,y,1)|x and y are real} is not subspace of R3.

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Q: Let W be the set of all vectors of the form Na subspace of R³ 0 2

A: Here given that  vector  [a 0 2]    We know that condition of subspace  w is zero vector belong…

Q: if it was M, N,L Subspaces of the vector space v on the field F Prove that (Mn N) + (M n L) c M n (N…

A: Given that M, N, L are subspace of a vector space V over the field F. We have to prove that M∩N +…

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

A: Let W1  and W2 be the subspaces of finite dimensional vector space V which satisfy dimW1=m  and…

Q: Let u, = , and uz =0 Note that u, and u, are orthogonal but that ug is not orthogonal to u, or uz.…

A: The solution is given as

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: . Let W be the subspace of R3 spanned by the vectors vj=(1,0,7), v2=(6,1,7), and V3=(14,-1,13). Find…

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Q: Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ...,…

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Q: Suppose that V is a vector space with basis B= {bi | i E I} and S is a subspace of V. Let {B1,... ,…

A: Given that V is a vector space with basis B=bi|i∈I and S is a subspace of V. Let B1,B2,B3,…,Bk be a…

Q: Let and I be subspaces of vector space V, let TE L(V). Q2) If U and W are invariant under T, prove…

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Q: Find the orthogonal projection of V = onto the subspace W of R' spanned by U1 = , U2 Uz = VW ||

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Q: a) Prove that W = {(a,b, 2a + b)} is a subspace of R. (more space on the next %3D if needed)

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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 V be a vector space and let S = {v,,…, Vn} be a set of vectors in V. Prove ... that the span of…

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

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

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 = R and W = {x|x = (x1, x2, 1)}, Is W a subspace of V?

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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.