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What are linear combinations of a column space? What equation to they satisfy? Why are they useful in leaner algebra?

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I'm not sure what you are trying to ask here. –  Robin Chapman Sep 29 '10 at 18:17

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up vote 4 down vote accepted

A linear combination of vectors, say ${ x_1, \ldots , x_n}$, of a vector space on some field $\mathbb{K}$ is $$\sum_{i = 1}^{n} \lambda_i x_i$$, where $\lambda_i \in \mathbb{K}$ and not all of them are zero.

These kind of combinations are important in linear algebra because you can define "linear independence" and "generators", which, if combined, give you a "basis" for the vector space (you can find all the definitions here: Wikipedia).

For example, ${x_1, \ldots , x_n }$ vectors are said to be independent, if $$\sum_{i = 1}^{n} \lambda_i x_i = 0 \Rightarrow \lambda_i = 0 \quad \forall i$$. This definition, as you can see, uses the concept of linear combination. On the same note, you can say that hose vectors generate the space $V$ if $\forall x \in V \quad x = \sum_{i = 1}^{n} \lambda_i x_i$ for appropriate $\lambda_i \in \mathbb{K}$. As you can see, also this definition uses the concept of linear combination.

As for the rest of your question, I can't really grasp its meaning.

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thank you very much. –  Rook Sep 29 '10 at 19:10
    
No problem! I think that if you read the wiki article you'll find out a lot more than I told you. –  Andy Sep 29 '10 at 19:13

The column space of a matrix is simply the space spanned by all linear combinations of the columns of the matrix. If $\rm \:A_i\:$ denotes the $\rm i$'th column of $\rm \:A\:$ then $\rm \ b = Ax = A_i\: x_1 + \cdots + A_n\: x_n$ has a solution iff $\rm\:b\:$ lies in the column space. Said in geometric language, the column space spans the image (range) of the linear map corresponding to the matrix $\rm\: A\:$.

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