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What is the column space of a matrix? What equation do they satisfy? Why are they useful in linear algebra?

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closed as too broad by Strants, sandwich, choco_addicted, TravisJ, pjs36 May 26 at 2:20

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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