Verify all vector space axioms under the given operations (x,y,z)+(x',y'z') = (x+x'+1,y+y'+2,z+z'+3) and k(x,y,z) = (kx,ky,kz)
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Q: 2. Let n₁,..., nk be pairwise relatively prime positive integers, and let n = n₁ ... nk. Recall we…
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Q: can both parts be written on paper please, if you can only do one then let it be part
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Q: 0881 144 (1)| +| (2) |log3 2 (1) = (2)
A: Properties are used.
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- Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector space?Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1 + x2 + 1, y1 + Y2+ 1) and k (*1, Y1) = (kx1, kyı) is not a vector space.What is the dimension of the vector space spanned by the vectors v1, v2? v1 = (6, -2, 4) and v2 = (14, 2, 8)
- Show that the vectors are linearly dependent or linearly independent. note: {2t^2 − 1, t + 1},P2:The space of all polynomials with a degree less than 2 and 2Let V=R^2 and let H be the subset of V of all points on the line 2x+3y=6. does H contain the zero vector V? is H closed under addition? is H closed under scalar multiplication?If (x1,y1) + (x2,y2) = (x1x2,y1y2) and c(x1,y1)= (cx1,cy1) are vector spaces, verify if both are a vector space property and identify the properties that it will fail. (Use the 8 theorems)
- Find a subset ofbthebvectors v1=(0,2,2,4), v2=(1,0,−1,−3), v3=(2,3,1,1) and v4=(−2,1,3,2)that forms abasis for the space spanned by these vectors. Explain clearly.Let S = {v1, v2, v3} be a set of linearly independent vectors in R^3. (a) Are the vectors in the set T1 = {v1, v1 + v3, v3} linearly independent? Show how you arrived at your answer. (b) Are the vectors in the set T2 = {v1 − v3, 3v1 + v2, v2 + 5v3} linearly independent? Show how you arrived at your answer.Let V₁, V2, V3 be the vectors in R³ defined by 18 ---D V₁ = -6 V2 = -14 V3 = 20 (a) is (V1, V2, Vs} linearly independent? Write all zeros if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of V₁, V₂, and V3 0 (c) Type the dimension of span {V1, V2, V3}: Note: You can earn partial credit on this problem. -25 -18] 0=v₁+√₂+vs (b) Is (v1, vs} linearly Independent? Write all zeros if it is, or if it is linearly dependent write the zero vector as a non-trivial (not all zero coefficients) linear combination of V₁ and V3. 0=v₁+v₁ V3
- 8. (a) Show that the vectors v1 = (1, 2, 3, 4), v2 = (0, 1, 0, – 1), and v3 = (1, 3, 3, 3), form a linearly dependent set in R*. (b) Express each vector as a linear combination of the other two.2- We can write the vector x = (19,–1, -) in R³ as a linear combination of the vectors x1 = (4,2, –1), x2 = (1,5,–1) and x3 = (2,-2,3), and the scalars are A:a, = -,a, = -,a, =-2 -,a2 = 5, az C: az = 5, az = -2,az B: a, = = 2 1 2 A В CDetermine if the list ( [111], [121], [011], [122] ) of vectors from vector space V = R^3 is linearly independent. If yes, show work, if no state a linear combination among the vectors of the list that adds up to a vector in the list. I am having trouble grasping the concept so if anyone can explain this problem I'd appreciate it.