Checking Subspace: Problem Solution

Thread starter Clandry
Start date Oct 27, 2012
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Subspace

In summary, "Checking Subspace: Problem Solution" is a method used in linear algebra to determine if a given set of vectors form a subspace within a larger vector space. It is important to check for subspaces because it helps in understanding the structure and properties of vector spaces, and in solving problems involving vector spaces and linear transformations. To check for subspaces, we need to verify three conditions: closure under addition, closure under scalar multiplication, and the existence of a zero vector. If one of these conditions is not met, then the set of vectors is not a subspace. An example of "Checking Subspace: Problem Solution" is when we have a vector space V and a set of vectors S, and we can check if

Oct 27, 2012

Clandry

Hi. I'm trying to check if my approach is right.

The problem is attached.

I need to check these:
1) 0 vector is in S
2) if U and V are in S then U+V is in S
3) if V is in S, then cV where c is a scalar is in S

The 1st condition is not satisfied right?
Since A*[0 0]^t=[0 0]^t≠[1 2]^t?

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Oct 27, 2012

hedipaldi

Right.

FAQ: Checking Subspace: Problem Solution

What is "Checking Subspace: Problem Solution"?

"Checking Subspace: Problem Solution" is a method used in linear algebra to determine if a given set of vectors form a subspace within a larger vector space.

2. Why is it important to check for subspaces?

Checking for subspaces is important because it allows us to understand the structure of vector spaces and their inherent properties. It also helps in solving problems involving vector spaces and linear transformations.

3. How do you check for subspaces?

To check for subspaces, we need to verify three conditions: closure under addition, closure under scalar multiplication, and the existence of a zero vector. If all three conditions are met, then the set of vectors is a subspace.

4. What if one of the conditions for subspaces is not met?

If one of the conditions for subspaces is not met, then the set of vectors is not a subspace. In this case, we cannot use the properties of subspaces to solve problems and we may need to find a different approach.

5. Can you provide an example of "Checking Subspace: Problem Solution"?

Sure, let's say we have a vector space V = {(x, y, z) | x, y, z are real numbers} and a set of vectors S = {(1, 2, 3), (2, 4, 6)}. We can check if S is a subspace of V by verifying the three conditions. Let's take closure under addition, (1, 2, 3) + (2, 4, 6) = (3, 6, 9) which is also in V. Similarly, for closure under scalar multiplication, k(1, 2, 3) = (k, 2k, 3k) which is also in V. Finally, the zero vector (0, 0, 0) is also present in S. Since all three conditions are met, S is a subspace of V.