Consider the vectors u_1 = (1, 1, 1, 1), u_2 = (0, 1, 1, 1), u_3 = (0, 0, 1, 1) and u_4 = (0, 0, 0, 1). Write down an arbitrary vector (a_1, a_2, a_3, a_4) in RR^4 as a linear combination of u_1, u_2, u_3 and u_4.

Quinlan7g 2022-09-17 Answered
Question: Consider the vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1) and u4 = (0, 0, 0, 1). Write down an arbitrary vector (a1, a2, a3, a4) ∈ R 4 as a linear combination of u1, u2, u3 and u4.
Can I just do ( a 1 a 2 a 3 a 4 ) = k u 1 + b u 2 + c u 3 + d u 4? Is that it? What is the question trying yo highlight with all those ones?
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Answers (1)

hampiova76
Answered 2022-09-18 Author has 5 answers
HINT
Notice that e 1 = u 1 u 2 , e 2 = u 2 u 3 , e 3 = u 3 u 4 and e 4 = u 4
Then you can express the given vector as the following linear combination:
( a 1 , a 2 , a 3 , a 4 ) = a 1 e 1 + a 2 e 2 + a 3 e 3 + a 4 e 4
Now it remains to make the corresponding substitutions.

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