(b) If V = U OW is a direct sum of subspaces U, W C V then any element v E V can be written uniquely as v = u +w with u E U and w E W. Show that t(v) =u defines a linear map t: V → V. (c) Show for the map n constructed in (b) that Im(t) U and Ker(t) = W. %3D

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

Q: If V3(R) is a vector space and W₁ = {(a,0, c): a, c = R} and W₂ = {(0,b,c): b, c = R} are two…

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Q: 3. Determine whether X1 W E R? : x² + x2 = | X2 is a subspace of R².

A:

Q: f W1 and W2 are subspace of a vector space V(F),  then show that W1+W2 is also a subspace of V(F)

A: Given that W1 and W2 are two subspace of a vector space V over a field F.

Q: Consider a linear map T: U → R² and uí, u₂ € U. Given that T (u) = (-3,6) and T ԱՉ mapped to (0, 0)…

A: Given: Tu1→=-3,6T ; Tu2→=5,-10T.We need to find the vector which is mapped to 0,0T.

Q: Show that the map T : R3 -→R? given by T (x1, X2, X3) = (x1 + x2, X3) is an open map.

A:

Q: Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x+ azx² + a3x³) = ao +…

A: For T to be an isomorphism we must check that it is one one and onto both. And we can check it by…

Q: a Let H = :a – 76 + 3c – d = 0}. First show that H is a subspace of M2×2. Then determine dim(H). d

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Q: Consider the the linear maps ƒ : R² → R² and g : R² → R² given by f((x, y)) = (x − y, 2 + y) and…

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Q: Find the closest point to y in the subspace W spanned by v₁ and v2. y= 3 -1 1 16 V1 11 1 -4 1 5 V2 =…

A:

Q: Determine whether W is a subspace of V

A: Given

Q: Let : R³ → R¹ be a linear map such that /(1,2)=(1,1,2), and /(2, 1) = (2,2,1). Then which equalitie…

A: Let V(F) and W(F) be two vector spaces over field F and f:V→W be a linear map. Then, a)…

Q: If V3(R) is a vector space and W₁ = {(a,0, c): a, c = R} and W₂ = {(0, b, c): b, c = R} are two…

A: Provided that V3 is three dimensional vector space over field R and V has two W1 and W2 subspace…

Q: Determine whether W is a subspace of V

A: Determine whether W is a subspace of V

Q: . Consider M₂ (R) as a vector space and its subspaces U₁= {A € M₂(R): A¹ = A} and U₂ = {A € M₂(R) :…

A:

Q: Let S = {x3, x² + 2x, x² + 1}. Is S is span(P2(x)) or not? Let T = {(2,-2,4), (3, -5,4), (0,1,1)}.…

A:

Q: Determine whether U is a subspace of R³. U = {(2r, –s²,t)lr,s, and t E R}

A: Vector Subspace Test: A non-empty subset W of a vector space V is a vector subspace of V if and only…

Q: 1. L: V3 V4 defined by L(a, b, c) = (a - b,b+c-d, a+b-2d, b-c-d) 2. L: V4 → V3 defined by L(a, b, c,…

A: According to Bartleby guidelines we can only solve 1 question or three subparts. So put other one…

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: Find the closest point to y in the subspace W spanned by v₁ and v₂. y = 5 - 1 1 18 V₁ 1 - 2 - 1 3 V2…

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Q: 5. Let U = {(u₁, U2, U3) E R³|u₁ + u2 - u3 = 0} and V = {(V1, V2, V3) E R³v₁ - 3v2 +5v3 = 0}. (a)…

A: (a)

Q: Describe all linear maps ƒ : R² → R² that are projections with image the subspace ((1,2)).

A: f: ℝ2→ℝ2Subspace =1, 2

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

A:

Q: Describe explicitly a linear transformation from V, (R) into V, (R) which has its range the subspace…

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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: Let f: R2 R³ be a linear map such that ƒ(1, 2) = (1,1,2), and ƒ(2, 1) = (2,2,1). Then which…

A: We are given that f is a linear map such that f(1,2)=(1,1,2) and f(2,1)=(2,2,1)

Q: If W₁ and W₂ are subspaces of V and dim W₁ (a) W₁ = W₂. (c) W₂ C W₁. dim W2, then (b) W₁ C W₂. (d)…

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: Let: R² R³ be a linear map such that /(1,2)=(1,1,2), and /(2,1)=(2,2,1). Then which equalitie are…

A: