(True or False) Let V be a vector space with dimension equal to 4. 1. Every subs
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Question
(True or False)Let V be a vector space with dimension equal to 4.
1. Every subspace of V has dimension at most4. (T / F)
2. There is a subspace of V with dimension4. (T / F)
3. Every set of vectors from V with at least 4 elements spans thewholespace. (T / F)
4. It is possible to find 5 linearly independent elements ofV. (T / F)
5. If a subset of V is linearly independent and spans V, then ithas 4elements. (T / F)
6. If a subset of V has 4 elements and spans V, then it cannot belinearly dependent. (T / F)
7. It is not possible to span V with only 3vectors. (T / F)
8. If a spanning set has at most as many elements as a linearlyindependent set, then the two sets have the same number ofelements. (T / F)
Explanation / Answer
Examples assume V = R4 1) True. Subspace dimensions could exist of 0, 1, 2, 3, or4 2) True. V is a subspace of itself with dimension 4 3) False. Consider the subspace spanned by {(1, 0, 0,0)T, (2, 0, 0, 0)T, (3, 0, 0, 0)T,(4, 0, 0, 0)T} does not span V. 4) False. A 4-dimensional vector space has exactly 4 linearlyindependent vectors. 5) True. 6) True. A 4-dimensional vector space has exactly 4 linearlyindependent vectors. 7) True. 8) Honestly, don't understand this one completely, but my guess isit's True.
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