3) Suppose V and W are finite-dimensional. Let v e V.Let E = {T € L(V, W)|Tv = 0}. (a) Show that E is a subspace of L(V, W). (b) Suppose v 0. What is dim E ?

Q: If W is a subspace in an inner product space V and V = W + W, prove that V = WW¹.

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Q: 1. Consider the following subspaces of R³: U = {(x,x,y): x,y ER} and W = {(x,-x, 2x): x, y ≤ R} (a)…

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Q: 3. Let W = {(x,y,z): x? + y? = z}. Is W a subspace of V = R3? %3D

A: We need to check if W is a subspace or not.

Q: 15. Let U and V be subspaces of a vector space W. Show that if W = UV, then UnV = {0}.

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Q: B 1 Find the dimension of the subspace V span{v1, v2, V3, v4} in R³ where vị = 2 1 V2 = , V3 = and…

A: We find dimension of the subspace V

Q: B 6. Which of the following is a subspace of R³? a. {(x,y,z)/x-3y-1} b. {(x,y,z)/x=3y+1} c.…

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Q: 6. Consider the subset S of R2, S = {(x, y) | x – y = 1}. Decide if S is a subspace of R

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Q: 11. Let W be the set of all vectors of the form 5 b + b 2 с where b and care arbitrary. Find vectors…

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Q: Let V = R, the real line. Let W be a subset of V, where W = {x € R|x ≥ 0}. Is W a subspace of V?…

A: Given that the vector space V=R , the real line  . The subset W of V is said to be a subspace of V…

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

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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: 8. Let V be a vector space over F and v₁, ₂, € V. Then (₁₁) = (a₁v₁ + + a₂v, la,,a,F) is a subspace…

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Q: 142 Q2. Let A = Determine a basis for each of the subspaces CS(A"), |3| 316 N(A), CS(A), and N(A").

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Q: Decide if the following subsets are: {(x,y,z) e R³: 2x + y - 3z = 1}. {(x, y, z) e R³: 2x = y + z}.…

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Q: If U is a subspace of V, show that T(U)= {T(u) u in U} is a subspace of W (called the image of U…

A: Let U be a subspace of V. We have to show that TU=Tu| u∈U is subspace of W.

Q: 13. Which of the following is a subspace of R4? Justify your answer. a) S = {(a,b, c, d) E R* | a…

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Q: S Let H be the set of all vectors of the form 3s 2s such that H = Span {v}. Why does this show that…

A: H is the set of all vectors of the form s3s2s.  s3s2s=s132 where s is arbitrary. Thus any vector in…

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: of R". SICSZ, Show that Juppose that s, and Sz are subspaces. With the thet property, dim (s.)s dim…

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Q: , dim W 5. Let U, W be subspaces of a vector space V. If dim U and Un W = {0}, what is dim (U+ W)?

A: Let V be a vector space over a field k and let U, W be finite-dimensional subspaces of V.  Then…

Q: . Let u₁ = (1, -2, -1), U₂ = (2, 2,-2), and v = = (0, 0, 1). v is not in the subspace W spanned by…

A: In this question, a vector that is orthogonal to a given vector is to be determined. A non-zero…

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: Q.4 Show that U {(x, -x) |x in R} is a subspace of R. %3D

A: To Prove U is a subspace of vector space V 1) 0∈U  If x, y∈U  and a∈V  2) Closed under vector…

Q: 22. Let f be a linear functional on a subspace S of a finite-dimensional inner product space V. Let…

A: Please find the answer in next step

Q: - If X is a vector space and if X= MN, where M and N are subspaces, prove that N is isomorphic to…

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Q: b.) show that IR³ - W₁ + W₂. t C.) Find a subspace V such that IR³ =W₁ U.

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Q: If S is a subspace in R that is spanned by (:) G)-G) then S- is spanned by: а. 1 and 1 1 1 and -1 С.…

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Q: 3. This problem is about set addition and subspaces. Let d = and dz = and V = span{d}, %3D U =…

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Q: D

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Q: (3) Is Mmxn(Q) is a vector subspace of Mmxn (R) over R?

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

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Q: Let V, W be a pair of subspaces of Rn and suppose that W ⊂ V . Prove step by step in detail that…

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Q: The population of a town grows at a rate proportional to the population present at time t. The…

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Q: a Q.8. Let the set H = :a² +b >1 b Is set H a vector space? Justify your answer.

A: Vector space always has a zero vector.

Q: 2) Let V=R²³: W= { x = (X₁₁ X ₂₁ X 3) ER ³²: 2x₁ - x₂ + 5*3=0}. Prove that W is a subspace and find…

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Q: Let W = { |xy <0}. Is W a subspace of R?? If so, use generic terms to demonstrate the necessary…

A: W is not a subspace of R2.