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In Exercises 7–10, show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for ℝ2 or ℝ3, respectively Then express x as a linear combination of the u’s.
8. u1 =
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Linear Algebra and Its Applications (5th Edition)
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- 6 Let b₁ =[2] b₂ = [4] 3 J CI 13 V= V -2 9 ²₁-[2] 2² =²²] =1² -2 and consider the bases for R² bz given by B = { b₁,b₂} and C = {0₁₂ (₂3 Find a) b) [V] b; part a) c) [V]c using the results from a) and b) the change-of-coordinates matrix from B to Carrow_forwardFind the change of basis matrix from B, to B2. B, - {[}: (:}, 5 - {(})-[}} -3| -1arrow_forwardWhat is the basic way to find a basis for all of the vectors x y z that satisfies the matrix equation of a 3 x 3 matrixarrow_forward
- If {æ, x3} is a basis of solutions of x²y" + axy' + by = 0 , then a = (А) —2 (В) 2 (C) -3 (D) 3 (E) None O A ов C D O Earrow_forwardB = (f1; f2; f3) is a basis for R2[r]. If the change of basis matrix from B to 1 2 3 Szalz] is (1 3 4 then f2 = 1 4 a) z b) 3 — г? c) -2+3r – r² d) 1+3r+ 4x² e) 2+3r + 4x²arrow_forward9) please be clear with your answer, thanks!arrow_forward
- 5. Explain why three linearly independent vectors u,v,w in R³ form a basis for R³. (Hint: Consider the 3 by 3 matrix A = [uvw]. Discuss the solution of the equation Ax=b for any b=R³.arrow_forwardLet B1 = {A1, A2, A3, A4} and B2 = {C1, C2. C3, C4} when : %3D A = G 0). 4 = 6 = (; }),cs- (? !),cs- (::)«-(:?). 1 0 0 C2 C3 C4 = 1. Prove that B, and B2 are two basis of M2 (R). 6 2 2. Find the coordinates of D ( ) in the basis B1. 5 3 3. Give the transition matrix Ps,--B,. 4. Using 3. deduce the coordinates of D in the basis B.arrow_forwardIn Exercises 7-10, show that {u₁, u₂} or {u₁, U2, U3} is an orthogonal basis for R² or R³, respectively. Then express x as a linear combination of the u's. 7. 6 --G--0-×-Q = and and x = 4 = 2 -3 9arrow_forward
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