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.
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- Let v1, v2, v3 be the vectors in R^3 defined by: v1=[0 5 8], v2=[12,-20,5], v3=[-12,15,-13] a). Is {v1,v2,v3} linearly independent? Write all zeroes 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. a). Is {v1,v2} linearly independent? Write all zeroes 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 v2. c). Write the dimension of span {v1,v2,v3}Determine 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.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
- 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)Let B={(1,-2,1), (4,-7,5), (5,-8,8)), and x=(-6,10,-7) Find [x]B Give your answer in the form (a,b,c) with no spacesShow 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 2
- One of the following vectors lies in span{ 2 + x +x², 3+x – 2x² } (A) 3 + 2x + 4x² (B) 3 + 2x + 5x² (C) 3 + 2x ++ 6x² (D) 3 + 2x + 7x² (E) 3 + 2x + 3x?If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. [a + 6b] where a and b are arbitrary real numbers. 3b 12) W is the set of all vectors of the form 4а - b -a a - 4b] 13) W is the set of all vectors of the form where a and b are arbitrary real numbers. 4а + b -a - b]Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.
- demonstrate whether it's a vector space3) S contains only the vectors (-2+x-x² - 2x³), (4 + x - x² + 2x³), (8 + 5x -5x²+2x³), (10 + 7x - 7x² + 2x³). The new linearly independent set with the same span would beFind 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.