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freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Linear Algebra Help Needed!

Hi this is the question that's troubling me:

Show the following

(a) If

is a spanning set for a vector space V and v is any vector in V, then

are linearly dependent.

(b) If

are linearly independent vectors in a vector space V, then

cannot span V.

Thanks in advance!

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Linear Algebra Help Needed!

By definition of a spanning set, any vector in V can be made by a linear combination of the vectors v_1, ..., v_n, including v.

That is, there exist scalars α_1, ..., α_n such that α_1v_1 + α_2v_2 + ... + α_nv_n = v.

Rearranging gives that α_1v_1 + α_2v_2 + ... + α_nv_n + (-1)v = 0.
That means that there is a linear combination of the vectors v_1, ..., v_n, v (with at least one of the scalars not zero) that adds to 0. Therefore, the set is linearly dependent.

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Assume that v_2, ..., v_n span V. Then, by the previous result, v_1, v_2, ..., v_n are a linearly dependent set of vectors. Contradiction; therefore v_2, ..., v_n do not span V.

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Why did the vector cross the road?
It wanted to be normal.