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Linear Algebra and Its Applications | 5th Edition | ISBN: 9780321982384 | Authors: David C. Lay; Steven R. Lay; Judi J. McDonald
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s u = (u1, … , un), v = (v1, … ,vn) and w = (w1, … , wn)

Chapter 1, Problem 33E

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QUESTION:

PROBLEM 33EUse the vectors u = (u1, … , un), v = (v1, … ,vn) and w = (w1, … , wn) to verify the following algebraic properties of ?n.

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QUESTION:

PROBLEM 33EUse the vectors u = (u1, … , un), v = (v1, … ,vn) and w = (w1, … , wn) to verify the following algebraic properties of ?n.

ANSWER:

Solution(a)Step 1 of 2Consider the vector u = ( u1,.........,un ) , v = ( v1,............, vn ) and w = ( w1,...........,wn ) be the vector of Rn.So, we have to prove (u + v) + w = u + (v + w)Now, (u + v) + w = (( u1,.........,un ) + ( v1,............, vn ))+( w1,...........,wn ) = ( (u1 + v1) +........ +(un + vn ) ) + ( w1,...........,wn ) = ( ( (u1 + v1) +w1) +........ + ( (un + vn ) + wn ) ) = ( ( u1 + v1 +w1) +........ + ( un + vn + wn ) ) = (u1 +( v1 +w1)) + ........ + (un +( vn + wn ) ) = ((u1 +.........+ un) +( (v1+............ vn ) +(w1 + …………. + wn ) ) ) = ( u1,.........,un ) + (( v1,............, vn )+( w1,...........,wn )) = u + (v + w)Hence, it is proved.(b)

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