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Test 2 1553
| Question | Answer |
|---|---|
| Define Linear Independence (1 tenant) Hint: Trivial Solution | A set of vectors {v1, v2, . . . , vp} in R^n is linearly independent if the vector equation x1v1 + x2v2 + · · · + xpvp = 0 has only the trivial solution x1 = x2 = · · · = xp = 0. The set {v1, v2, . . . , vp} is linearly dependent otherwise. |
| Connection between pivots and linear independence and how do we go about doing this? | The vectors v1, ...vp are linearly independent only if the matrix with columns v1,...vp has a pivot in each column. Solving the matrix equation Ax = 0 will either verify cols of A are linearly independent, or show linear dependence relation. |
| Linear dependence theorem can be stated in two ways and flipped for linear independence hint: Span | A set of vectors {v1, .. vp} is linearly dependent only if one of the vectors is in the span of another. or A set of vectors {v1, v2, . . . , vp} is linearly dependent if and only if you can remove one of the vectors without shrinking the span. |
| Are there four vectors u, v, w, x in R 3 which are linearly dependent, but such that u is not a linear combination of v, w, x? If so, draw a picture; if not, give an argument. | Yes: actually the pictures on the previous slides provide such an example. Linear dependence of {v1, . . . , vp} means some vi is a linear combination of the others, not any. |
Created by:
molatunji
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