Practice question · Select all that apply
V is a vector space of dimension 4. Select every statement that MUST be true.
Hints
- Once the dimension is known, one of the two basis conditions implies the other, but only for the right count.
- Try four copies of the same vector, and try three vectors in a four-dimensional space.
Show the answer
- B. Any 5 vectors in V are linearly dependent
- C. Every basis of V has exactly 4 vectors
- D. Any 4 linearly independent vectors in V form a basis
Why
All bases share the dimension count, and in an n-dimensional space n independent vectors are automatically spanning. But four arbitrary vectors could be four copies of one vector, and three vectors can never reach a four-dimensional space. Five vectors exceed the dimension and must be dependent.
Practise Basis and Dimension
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