can you solve all qn with explanation linear algebra Math SE -Berkeley City Coll

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can you solve all qn with explanation linear algebra

Math SE -Berkeley City College, Fall '17 S. McDougal Quiz 1 Name: There are 5 problems worth 4 points each. Be sure to show work/justification as appropriate. You may not use calculators or notes. You have 12 minutes to complete the quiz. Given A = (1-1) and B = (4,2), find AB 2. Describe-in words or pictures-all vectorsthat are orthogonal to u -21 , determine whether the angle between them is acute Given the vectors u =1 , v (ie. les than ), obtuse, or a right angle 3. te For problems 4 and 5, indicate whether it's True or False. (No need to justify or show work.) 5. If u is orthogonal to both v and w, tben 4, 2r 1 doesn't have a solution in Z. u is orthogonal to v + w. rde Extra Credit (2 points) Given the vectors u = |-4/51 ,V-BI. find prolu(v)

Explanation / Answer

1) AB = B -A = (4,2) -(1,-1) = (3,3)
2)orthogonal means
u.v = 0
(x,y).(3,1) = 0
3x+ y = 0
y = -3x
or (x,-3x) = x(1,-3)

3) u.v = 0 right angle > 0 acute , < 0 then obtuse
here u.v = 2*1 -1*(-2) +1*(-1) = 2 + 2 -1 = 3 > 0
hence acute

4)Z4 has 4 elements 0,1,2,3
2x = 1
2*0=0 = 0   (in Z4)
2*1=2 = 2    (in Z4)
2*2=4 = 0   (in Z4)
2*3 = 6 = 2   (in Z4)
hence true

5) true
u .v = 0
u.w = 0
u.(v+ w) = u.v + u.w = 0+0 =0
hence u is orthogonal to v+w

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