Basis of sum/union of subspaces

In summary, to find the basis of W1 + W2 and W1 intersect W2, we can use the equation Dim(A) + Dim(B) = Dim(A intersect B) + Dim(A + B) and the given matrices W1 and W2. We can also use the vectors \alpha and \beta to find the dimensions of each subspace. Additionally, we can see that the vector \left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right) is in both W1 and W2, so it is also in the intersection. To find the other vector in the basis of the intersection, we can consider the equation for W1 + W
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Homework Statement


I need to find the basis of W1 + W2 and W1 intersect W2

(It's part of a larger homework problem that I know how to do, but I am stuck on the trivial step...per usual)

[tex]W_1 = \left(\begin{array}{c c} x & -x \\ y & z \end{array}\right)[/tex]

[tex]W_2 = \left(\begin{array}{c c} a & b \\ -a & c \end{array}\right)[/tex]


Homework Equations


Dim(A) + Dim(B) = Dim(A intersect B) + Dim(A + B)


The Attempt at a Solution



[tex]\alpha \in W_1, \alpha = x \left(\begin{array}{c c} 1 & -1 \\ 0 & 0 \end{array}\right) + y \left(\begin{array}{c c} 0 & 0 \\ 1 & 0 \end{array}\right) + z \left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)[/tex]

[tex]\beta \in W_2, \beta = a \left(\begin{array}{c c} 1 & 0 \\ -1& 0 \end{array}\right) + b \left(\begin{array}{c c} 0 & 1 \\ 0 & 0 \end{array}\right) + c \left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)[/tex]

So the dimensions of each subspace of those subspaces is 3.

Obviously, [tex]\left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)[/tex] is in both sets (and thus in the intersection).

I'm having a hard time finding the other vector in the basis.

Is [tex]\left(\begin{array}{c c} 1 & -1 \\ -1 & 0 \end{array}\right)[/tex] in the basis set of vectors of the intersection?
 
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  • #2
consider W_1+W_2.

you'll get

[tex]W_1+W_2 = \left(\begin{array}{c c} x+a & -x+b \\ y-a & z+c \end{array}\right)[/tex]

what would a basis for that be? and then its dimension?

also, keep in mind the equation you posted, and think about what you said about [tex]
\left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)
[/tex] being in the intersection.

what exactly does the intersection look like?
 

Related to Basis of sum/union of subspaces

What is the definition of basis of sum/union of subspaces?

The basis of sum/union of subspaces refers to the set of vectors that can be used to span the entire space formed by the sum or union of two or more subspaces.

How do you find the basis of sum/union of subspaces?

To find the basis of sum/union of subspaces, you can use the following steps:

  • Find the basis of each individual subspace.
  • Combine all the basis vectors from each subspace to form a new set of vectors.
  • Eliminate any duplicate vectors from the new set.
  • If the new set spans the entire space, then it is the basis of the sum/union of subspaces.
  • If the new set does not span the entire space, then you may need to add more vectors to the set to form the basis.

What is the significance of the basis of sum/union of subspaces in linear algebra?

The basis of sum/union of subspaces is important in linear algebra because it allows us to understand the structure of a space formed by the combination of multiple subspaces. It also helps us to find a set of vectors that can be used to span the entire space, which is useful in solving problems and performing calculations.

Can the basis of sum/union of subspaces change?

Yes, the basis of sum/union of subspaces can change depending on the subspaces that are being combined. If the subspaces are different, then the basis may change. However, if the subspaces are the same, then the basis will remain unchanged.

What is the relationship between the dimension of the sum/union of subspaces and the dimensions of the individual subspaces?

The dimension of the sum/union of subspaces is equal to the sum of the dimensions of the individual subspaces, minus the dimension of their intersection. In other words, if we have two subspaces with dimensions m and n, then the dimension of their sum/union will be m + n if the subspaces are disjoint (have no common vectors), or m + n - k if they have k common vectors.

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