Questions about Null Spaces...?

2008-12-10 12:53 pm

I ma writing a short essay on null spaces..can anyone find the error in my essay?

The vectors V1,V2,...,Vn are called linearly dependent if each of them can be written as a linear combination of the others. If V1,V2,...,Vn span the null space of a matrix and are linearly dependent, we can leave out some of the vectors such that the remaining vectors still span the null space. This is convenient because we can then describe the null space with fewer vectors V1,V2,....,Vn are linearly independent, i.e. if the vector equation

lambda1 V1 + lambda2 V2 +.....+lambda n Vn = 0

is solved by lambda1=lambda2=...=lambda n=0. If V1,V2,...,Vn span the null space of a matrix and are linearly independent, no smaller set of vectors can be found that spans the null space.
We say in this case that the vectors V1.V2,..,Vn are a basis of the null space.

回答 (1)

用戶8745

2008-12-10 8:58 pm

✔ 最佳答案

The main error is the statement

The vectors V1,V2,...,Vn are called linearly dependent if each of them can be written as a linear combination of the others.

Here, "each of them" should be "some of them", or more precisely,
"at least one of them".

(For example, consider (1,0), (0,1), (0,1), the first vector is not a linear combination of the other 2.)

Moreover, the statement

is solved by lambda1=lambda2=...=lambda n=0.

is not very precise. To improve it, replace the phrase "is solve by" by
"has unique solution"

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