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chang asked in 科學及數學數學 · 1 decade ago

Linear algebra part 1~urgent!!

Which of the following statements are true and which false? (If false, then give a counter example. If true, then give a brief explanation of at most one sentence)

i. A sequence of n linearly dependent vectors in an n-dimensional vector space cannot span the space.

ii. Any sequence if n+1 distinct non-zero vectors in an n-dimensional vector space must span the space.

iii. If T: U -> V is a linear map, then Rank (T) = [ dim(U) + dim (V) ] /2

1 Answer

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  • 志仁
    Lv 6
    1 decade ago
    Favorite Answer

    i) Taking R3 as an example, we have 3 linearly dependent vectors, i, 2i and 3i, it is obvious that these 3 vectors cannot span the space. For an n-dimensional vector space, we need n linearly independent vectors to span the space.

    ii) Taking R3 as an example, we have 4 vectors, i, i+j, 2i+j, and 10i-j. they are distinct non-zero vectors but cannot span the space.

    Any sequence if n+1 distinct non-zero vectors in an n-dimensional vector space MAY span the space.

    iii) For a linear map, Rank (T) <= min[ dim(U),dim (V) ], so the statement is false.

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