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- Illustrate with an example and a diagram why a+b=6+a¯ for all vectors a and ba. Write the vector (-4,-8, 6) as a linear combination of a₁ (1, -3, -2), a₂ = (-5,–2,5) and ẩ3 = (−1,2,3). Express your answer in terms of the named vectors. Your answer should be in the form 4ả₁ + 5ả₂ + 6ẩ3, which would be entered as 4a1 + 5a2 + 6a3. (-4,-8, 6) = -3a1+a2+2a3 b. Represent the vector (-4,-8,6) in terms of the ordered basis = {(1, −3,−2), (-5, -2,5),(-1,2,3)}. Your answer should be a vector of the general form . [(-4,-8,6)] =a, b and c are three vectors in the third dimension with the following properties: a × b does not equal zero. a × b = b × c = c × a. Show that a + b + c = 0. Show that a + b + c = 0.
- 6. Write DÉ as a linear combination of the vectors i and j a) D (9,-6) E (-7, 2) first write in component form, then write as linear combination b) D (-3, 5.7) E (6,-8.1) 10+N1Give the vector result of a + b + c as a sum of its i and j components. [Give an answer using the letters i and j, such as 15i+ 29 j.]
- Determine the vector 8a – 7b if a = -3i + 2j – 4k and b = 5i + 6j + 2k. 8a – 7b = help (vectors)Q2. ()- Work out a - 2b as a column vector.(Left nullspace) Add the extra column b and reduce A to echelon form: 1 2 3 b₁ 1 2 3 b₁ [A b] = 4 5 6 b2 → 0-3-6 b2-4b1 789 63 0 00 b3 - 252 + b₁ A combination of the rows of A has produced the zero row. What combination is it? (Look at b3 -2b2 + b₁ on the right side.) Which vectors are in the nullspace of AT and which vectors are in the nullspace of A?