VeeEight said:Let W be the intersection of W1 and W2. Is the vector space W closed under addition and scalar multiplication?
Well, if not, you're going to have a tough time proving it.mlarson9000 said:Wouldn't it have to be?
Take a couple of arbitrary vectors u1 and u2 in W, and show that their sum is also in W. Then take an arbitrary scalar s, and show that su1 is in W. That's how you would do it it "math speak."mlarson9000 said:If W is in both W1 and W2, which are both subspaces and therefore closed under addition and scalar multiplication, wouldn't W be also?
mlarson9000 said:If so, I still don't know exactly how to say that in Math speak.
Suppose u and v are in W. Then u and v are both is W1 and, since W1 is a subspace, u+ v is in W1. Also u and v are both in W2 ...mlarson9000 said:Wouldn't it have to be? If W is in both W1 and W2, which are both subspaces and therefore closed under addition and scalar multiplication, wouldn't W be also? If so, I still don't know exactly how to say that in Math speak.
The Subspace Intersection Problem is a mathematical problem that involves finding the intersection between two subspaces in a vector space. In other words, it is the process of determining if two subspaces share any common vectors.
Proving W1 and W2 in R^n is important because it helps us understand the relationship between two subspaces and how they interact with each other. It also allows us to determine if the two subspaces are independent or not.
To prove W1 and W2 in R^n, you can use the method of contradiction. Assume that the two subspaces share a common vector and then show that this leads to a contradiction. This proves that the two subspaces do not share any common vectors and are therefore independent.
Yes, the Subspace Intersection Problem can be solved in any vector space. However, the process and methods for proving W1 and W2 may vary depending on the dimensionality and properties of the vector space.
The Subspace Intersection Problem has many practical applications, such as in image processing, signal processing, and data compression. It is also used in machine learning and computer vision to determine the relationship between different data sets and identify common features.