Suppose A, B, C are vectors, O is the zero vector and A - B - C = 0. Which of th

Question

Suppose A, B, C are vectors, O is the zero vector and A - B - C = 0. Which of the following sets of vectors are certain to be linearly dependent? {-3A - 4B, -3A -4B} {5A - B - 5C, -3A- 4B, A - B- 5C} {-3A - 4B, A - B - 5C} {A - B - C, 2A - B - C, B} You answered:;;(-3A - 4B, A - B - 5C) Your answer part 1 is incorrect; later parts (if any) were not checked. Questions or comments concerning your last submission of Question 1 (code: 6) in the box below will be sent to your instructor with your next submission. Make sure you submit before leaving this session.

Explanation / Answer

(1) {-3A-4B, -3A-4B} is linearly dependent because both vectors are eqal to each other

(2) {5A-B-5C, -3A-4B, A-B-5C}

Assume, r (5A -B -5C) + s (-3A - 4B) + t (A - B - 5C) = 0

A (5r +-3s +t) +B (-r-4s-t) + C (-5r-5t) = 0

Now we have A-B-C=0 => A= B + C

(B+C)(5r-3s+t)+B (-r-4s-t) + C (-5r-5t) = 0

=> B (4r-7s) + C (-3s-4t) = 0

=> 4r-7s = 0=> r = (7/4)s

and -3s-4t=0 => s = (-4/3)t

Hence we have r = (-7/3)t, s=(-4/3)t, t

And so here r,s and t are non zero scalars and hence {5A-B-5C, -3A-4B, A-B-5C} is linearly dependent

(3) {-3A-4B, A-B-5C}

Suppose r(3A-4B) + s(A-B-5C) = 0

A(3r+s) + B(-4r-s) + C(-5s) = 0

A = B + C

=> (B+C)(3r+s) + B(-4r-s) + C(-5s) = 0

=> B (-r) + C(3r-4s) = 0

=> r = 0 and 3r=4s => s=0

Hence r=0, s=0

So the given set is not linearly dependent

(4) {A-B-C,2A-B-C,B}

Now we are given A-B-C =0 and so the given set of vectors is equivalent to:

{0, 2A-B-C,B}

Any set that contains 0 is always linealry dependent and so this is linearly dependendt

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