Practice question · Select all that apply
The set {u, v, w} is linearly DEPENDENT. Select every statement that must then be true.
Hints
- Two of these are just restatements of the definition in different language.
- Recall the set {(1,1,0), (0,1,1), (1,2,1)} from earlier in this lesson.
Show the answer
- A. At least one of the three is a linear combination of the other two
- C. The homogeneous system with u, v, w as columns has a nonzero solution
- D. Some combination of u, v, w with not all coefficients zero equals the zero vector
Why
Dependence means a nontrivial vanishing combination exists, which is the same as the homogeneous system having a nonzero solution, and solving that relation for the vector with a nonzero coefficient exhibits it as a combination of the others. But no vector need be a MULTIPLE of another, the sum-of-two-others example shows why.
Practise Linear Independence
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