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Let us say that there is a linear combination $c_1\mathbb{x_1} + c_2\mathbb{x_2} + c_3\mathbb{x_3} + c_4\mathbb{x_4}$.

where $x_k$ is a $n \times 1$ matrix.

How do I figure out the basis of the space generated by this linear combination formally?

For example, if $\mathbb{x_1} = \begin{bmatrix}0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \mathbb{x_2}= \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1\end{bmatrix}, \mathbb{x_3} = \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \mathbb{x_4} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}$, how do I figure out the space of the basis of the linear combination of these matrices?

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What do you mean by 'basis of the linear combination'? Are you looking for a basis for the space generated by the set $\{x_1,\ldots,x_4\}$? –  Sigur Aug 29 '12 at 0:11
    
@Sigur yes, I am. –  Halo Dotcom Aug 29 '12 at 0:15
    
First of all, any set with zero vector is linearly dependent, so can not be a basis. So, you have to search among the other 3 vectors the maximal linearly independent set. –  Sigur Aug 29 '12 at 0:17
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you don't need $x_1$ and $x_4=x_2+x_3$ so a spanning set is $x_2$ and $x_3$. also $x_2$ and $x_3$ are linearly independent (easy) so the basis is $x_2, x_3$ –  i. m. soloveichik Aug 29 '12 at 0:18
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You need to read a chapter from a book or something. Alternatively, since we live in digital times, you can watch Gilbert Strang's lectures on MIT OCW (Search Youtube for: Gilbert Strang linear algebra). –  user2468 Aug 29 '12 at 1:58
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2 Answers

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All you have to do is row reduce! Put everything into one matrix, and get rid of as many rows as you can.

Logically, in a basis, first of all there cannot be the zero vector. Secondly, no vector in a basis can be a linear combination of any other vectors. In your example, $X_4 = X_3 + X_2$, so you can leave out $X_4$. You are left with your basis: $X_2, X_3$

If you want to row reduce with a more complex question, you have the following matrix: $\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{array} \right]$

Move the zero row to the bottom for convenience.

$\left[ \begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array} \right]$

Now subtract the first row + the second row from the third row.

$\left[ \begin{array}{cccc} 0 & 0 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$

We cannot row reduce anymore. Your answer is the rows that are not completely 0's. The first row, which maps to $X_2$, and the second row, which maps to $X_3$, is your basis. Remember, for the basis, you should use the original vectors, so the original $X_2$ and $X_3$ (coincidentally they are the same this time).

So your basis is $X_2, X_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1\end{bmatrix},\begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}$

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That is, of course, a basis. –  Gerry Myerson Aug 29 '12 at 1:19
    
@GerryMyerson What do you mean? Like other basis can be created through scalar multiplication of those two vectors right? or is there something else? –  mathguy Aug 29 '12 at 1:31
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When you write "your basis is..." there's a possibility someone will take this to mean that's the only possible basis for the space in question. But in fact any two vectors $ax_2+bx_3,cx_2+dx_3$ will do, so long as $ad-bc\ne0$. I'm sure you know this, and I wasn't trying to say that what you wrote was wrong - I was just adding a little bit, lest anyone come away with the wrong impression. –  Gerry Myerson Aug 29 '12 at 3:09
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You mean a basis for the linear transformation. The linear combination you mentioned induces a linear operator $T:\mathbb{R}^n\to\mathbb{R}^m$ (in your case $m=1$). Then,

$$ T(\mathbf y)=\mathbf{X}\cdot\mathbf{y} $$

where $\mathbf{X}\in M_{m\times n}(\mathbb{R})$ and $\mathbf{y}\in \mathbb{R}^n$

$$ \mathbf X=\left[ {\begin{array}{cccc} x_1&x_2&\cdots&x_n \end{array} } \right] $$

The space generated by these vectors ($\{\mathbf{x}_i\}_i$) is their span. Formally put, it is the image of $T$:

$$ \operatorname{span}\{c_i\}_{i=1,2,\ldots,n}=\operatorname{im}T $$

Recall the definition of the image of $T$. It is:

$$ \operatorname{im}T = \left\{ \mathbf z| \mathbf z=T(\mathbf y); \mathbf y\in \mathbb{R}^n \right\} $$

which sometimes appears as $T(\mathbb{R}^n)$. The image of $T$ is a linear subspace of $\mathbb{R}^m$. The dimension of the image of $T$ equals the rank of $T$, that is the maximum number of linearly independent columns of $C$. The rank-nullity theorem is a quite important result when working with dimensions of subspaces ($\operatorname{dim}\operatorname{im}T+\operatorname{dim}\operatorname{ker}T=n$.)

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If I well understood, he is looking for a basis for the space generated by those 4 vectors. –  Sigur Aug 29 '12 at 0:18
    
@Sigur I updated my answer. I will refine it a bit more, wait... –  Pantelis Sopasakis Aug 29 '12 at 0:23
    
The image of $T$ is the column space of the matrix representing $T$, so if you want the image of $T(x)=Ax$ to be the span of the given vectors $x_1,x_2,x_3,x_4$ then you had better let $A$ be the matrix whose columns are those 4 vectors. Maybe that's what you're saying, but you lost me when you switched from the $x_i$ to the $c_i$. –  Gerry Myerson Aug 29 '12 at 1:24
    
@GerryMyerson You are right. I updated my reply to follow the same notation. –  Pantelis Sopasakis Aug 29 '12 at 1:48
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