 # 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 $\left(\begin{array}{c}a1\\ a2\\ a3\\ a4\end{array}\right)=k\cdot u1\phantom{\rule{mediummathspace}{0ex}}+\phantom{\rule{mediummathspace}{0ex}}b\cdot u2\phantom{\rule{mediummathspace}{0ex}}+\phantom{\rule{mediummathspace}{0ex}}c\cdot u3+d\cdot u4$? Is that it? What is the question trying yo highlight with all those ones?
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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:
$\begin{array}{r}\left({a}_{1},{a}_{2},{a}_{3},{a}_{4}\right)={a}_{1}{e}_{1}+{a}_{2}{e}_{2}+{a}_{3}{e}_{3}+{a}_{4}{e}_{4}\end{array}$
Now it remains to make the corresponding substitutions.

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