Finding Basis for Intersection of Subspaces of $\Bbb R^n$

[Evgeny.Makarov]

I am looking for a method of finding a basis of the union and intersection of two subspaces of $\Bbb R^n$. My question is primarily about the intersection. Suppose that the basis of $L_1$ is $\mathcal{A}=(a_1,\dots,a_k)$ and the basis of $L_2$ is $\mathcal{B}=(b_1,\dots,b_l)$. Then $v\in L_1\cap L_2$ iff there exist $x_1,\dots,x_k$ and $y_1,\dots y_l$ such that
\[
x_1a_1+\dots+x_ka_k=y_1b_1+\dots+y_lb_l=v
\]
So we form a matrix where $\mathcal{A}$ and $\mathcal{B}$ are columns and reduce it to echelon form. The columns with pivots (suppose there are $s$ of them) correspond to vectors from $\mathcal{A},\mathcal{B}$ that make up the basis of $L_1\cup L_2$. The columns without pivots (there are $d=k+l-s$ of them) correspond to free variables. So we assign $d$ linearly independent sets of values to free variables, and they determine all of $x_1,\dots,x_k,y_1,\dots,y_l$. For each such solution, $\sum_{i=1}^l y_ib_i$ is in the basis of $L_1\cap L_2$.

So far so good. I would also like to know how to save myself some calculations when the given vectors $\mathcal{A},\mathcal{B}$ are not necessarily linearly independent. We can still reduce the matrix of these vectors as columns to echelon form. Then columns with pivots from the $\mathcal{A}$ part correspond to the basis of $L_1$. But concerning columns without pivots from the $\mathcal{B}$ part we don't know whether they belong to the basis of $L_2$. They correspond to free variables only when they do also correspond to vectors from the basis of $L_2$, and this means that we have to find the basis of $L_2$ first. Is it correct?

For example, suppose that $n=k=l=3$ and when we write vectors as columns and reduce the matrix to echelon form we get
\[
\begin{pmatrix}
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\
& \cdot & \cdot & \cdot & \cdot & \cdot\\
&&& \cdot & \cdot & \cdot
\end{pmatrix}
\]
where nonzero entries are denoted by $\cdot$. This means that $a_1,a_2$ form the basis of $L_1$ and $a_1,a_2,b_1$ form the basis of $L_1\cup L_2$. But we don't know if $b_2$ and $b_3$ are in the basis of $L_2$ along with $b_1$. If they are, then we set $y_2=1,y_3=0$, which determines $y_1$, and then we set $y_2=0,y_3=1$, which gives another $y_1'$. In this case, the basis of $L_1\cap L_2$ is $(y_1b_1+b_2,y_1'b_1+b_3)$. If, on the other hand, the basis of $L_2$ is $(b_1,b_2)$, then the free variable is only $y_2$. Setting $y_2=1$ determines $y_1$ and the basis of $L_1\cap L_2$ is $(y_1b_1+b_2)$.

So, the question is: do I need to find the basis of $L_2$ before reducing the whole matrix?

 

[jvicens]

I would recommend taking a step back and considering the problem from a more general perspective. Rather than focusing on specific bases and matrices, let's think about the underlying concepts and properties.

First, let's define the union and intersection of two subspaces. The union of two subspaces $L_1$ and $L_2$ is the set of all vectors that can be expressed as a linear combination of vectors from $L_1$ or $L_2$. The intersection of $L_1$ and $L_2$ is the set of all vectors that can be expressed as a linear combination of vectors from both $L_1$ and $L_2$.

Now, let's think about the basis of the union and intersection. The basis of the union of $L_1$ and $L_2$ would be a set of vectors that span the entire space of $L_1 \cup L_2$. This set of vectors would need to include all the vectors from both the basis of $L_1$ and the basis of $L_2$. In other words, it would be a combination of the two bases.

For the intersection, the basis would be a set of vectors that span the space of $L_1 \cap L_2$. This set of vectors would need to include all the vectors that are common to both $L_1$ and $L_2$. In other words, it would be the intersection of the two bases.

Now, let's think about the case where the given vectors are not necessarily linearly independent. In this case, the basis of $L_1$ and $L_2$ would not be unique, as there would be more than one way to express the same vector. However, the basis of the union and intersection would still have the same properties as mentioned above. We would still need to include all the vectors from both bases for the union, and only the common vectors for the intersection.

In summary, instead of focusing on specific bases and matrices, I would recommend thinking about the underlying concepts and properties of the union and intersection of subspaces. This approach may help save you some calculations and provide a more general understanding of the problem.