Let V R. For u, v E V and a € R define vector addition by uv : u+v+2 and scalar multiplication by au : au +2a - 2. It can be shown that (V,B, ) is a vector space over Find the following: the scalar field R. the sum: -9-9= the scalar multiple: -30-9- the zero vector: Oy = the additive inverse of a: Bx =

Q: Find vector c × (a + 3b), where a, b, and c are given below. ijk i j k -1 a = 509,b=0 0 1 0 -9i+27i…

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Q: The scalar product 0v is 0.

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Q: Evaluate the Dot Product u * (v + w). If : u = (-5, 7) v = (1, 6) w = (-11, 2)

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Q: (d) s(v+ww) × (rv)

A: Given that u→,v→,w→∈ℝ3 and r ,s ∈ℝ.

Q: Let x 4 || - N and y = Find the vector v = 3x - 4y and its additive inverse. 18 [2] 4

A: The additive inverse of a vector v = <a , b> is  - v = <-a , -b>

Q: 3. Prove that Ov=0 for every vector v in a vector space.

A: We have to prove that 0v = 0 for every vector v in a vector space.

Q: Select the option that gives the vector product of i+k and i + j Select one: 01 Oi+j-k Oi-j-k…

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Q: Which of these is a valid expression in index notation which outputs a vector? A d,F B © c) E F. F…

A: Given: Objective: Identify the valid expression in index notation that outputs a vector.

Q: Describe the additive inverse of a vector in the vector space.R3

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Q: Find the scalar triple product u (vxw). u. (vxw) = u = (-6,1,3), v = (3,6, - 8), w = (-4,3,2)

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Q: Find u · (v x w). This quantity is called the triple scalar product of u, v, and w. u = (4, 4, 4), v…

A: = i(5·0 - 0·(-1)) - j (1·0 - 0·0) + k (1·(-1) - 5·0) = i (0 - 0) - j (0 - 0) + k (-1…

Q: Let T: R³ R³ be the transformation that reflects each vector x = x₂ = 0 to the vector T(): = x1 -x2…

A: x→=x1x2x3Tx→=x1-x2x3

Q: 2. Prove that every element in a vector space has a unique additive inverse.

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Q: Calculate the scalar triple product, (ux v). w, ofu, v, and w where u = -141-43, v = 10i-20 + 15k,…

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Q: Find u ∙ (v × w). This quantity is called the triple scalar product of u, v, and w.u = (2, 0, 1), v…

A: Given, the three vectors u = (2, 0, 1), v = (0, 3, 0), w = (0, 0, 1). And we have to find the triple…

Q: Find the length of the projection of [−8 −7]onto the vector [−7 5]   Its written vertically…

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Q: 5. Prove that (-1)v=-v for every vector v in a vector space.

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Q: Find a vector orthogonal to both v and W. v = 3i + 2j + k, w = -5i - 5j - k 3i - 2j - 5k -3i + 2j -…

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Q: matrix representation of the linear operator on I

A: We have to find out the matrix representation of the linear operator on ℝ2. It is given that the…

Q: Let z = 6. and y Find the vectori = 47 - 7j and its additive inverse.

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Q: Which of these operators produces a scalar when it acts on a vector? A B D Ⓡ u. V V D2 V. Vx

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Q: Do 4

A: Part(4):- To show that:- a0 = 0 for every scalar "a" in a vector space. Here, "0" denotes zero…

Q: Let u = [1, -6, 3] and v= [-3, 3, 0]T. Find the vector w = 3u - 2v and its additive inverse. W= W=…

A: The given vectors and .We have to find the vectors and find its inverse.