Linear Algebra - Vectorial Subespaces

[encomes]

Homework Statement

It's about Linear Algebra and vector spaces. I've tried it but i can't get the solution...:

In C$$^{5}$$, the vectorial subespace U generated for (1,2,-1,-1,2), (0,2,-1,0,-2), (00,2,-1,0) and the vectorial subespace V generated for (3,3,0,-5,2), (1,1,0,-3,2), (1,1,0,1,-2).

I have to find a base and the dimension of the vectorial subespaces U, V, U+V and U $$\cap$$V. Do U and V define the same vectorial subespace?

Homework Equations

None.

The Attempt at a Solution

I have to find a base and the dimension of the vectorial subespaces U, V, U+V and U $$\cap$$V. Do U and V define the same vectorial subespace?

Thank you!

 

[anathema]

Hello! It seems like you are working with two vector subspaces in C^5, U and V. In order to find the dimension and bases for these subspaces, you can use the following steps:

1. To find the dimension of a vector subspace, you can use the rank-nullity theorem, which states that the dimension of a subspace is equal to the rank of its matrix representation minus the nullity (dimension of the null space). In this case, you can represent U and V as matrices and find their ranks and nullities to determine their dimensions.

2. To find a basis for a vector subspace, you can use the row reduction method to put the matrix representation of the subspace in reduced row echelon form. The columns that contain pivots in the reduced matrix will form a basis for the subspace.

3. To find the dimension and basis for U+V, you can use the properties of vector subspaces. Since U+V is the set of all vectors that can be written as a sum of a vector in U and a vector in V, you can find a basis for U+V by combining the bases for U and V. The dimension of U+V will be the sum of the dimensions of U and V.

4. To find the dimension and basis for U ∩ V, you can use the fact that U ∩ V is the set of all vectors that are in both U and V. Therefore, the basis for U ∩ V will be the intersection of the bases for U and V, and the dimension will be the number of vectors in this basis.

5. To determine if U and V define the same vector subspace, you can compare their bases and dimensions. If they have the same basis and dimension, then they define the same vector subspace.

I hope this helps! Let me know if you have any further questions or need clarification. Good luck!

 

[tomcue]

To find a base for a vector space, we need to find a set of linearly independent vectors that span the space. For U, we can see that the vectors (1,2,-1,-1,2), (0,2,-1,0,-2), and (0,0,2,-1,0) are linearly independent and span U. Therefore, a base for U is {(1,2,-1,-1,2), (0,2,-1,0,-2), (0,0,2,-1,0)} and the dimension of U is 3.

Similarly, for V, the vectors (3,3,0,-5,2), (1,1,0,-3,2), and (1,1,0,1,-2) are linearly independent and span V. So a base for V is {(3,3,0,-5,2), (1,1,0,-3,2), (1,1,0,1,-2)} and the dimension of V is 3.

For U+V, we can see that all the vectors in U and V are linearly independent, so a base for U+V would be the union of the bases for U and V. Therefore, a base for U+V is {(1,2,-1,-1,2), (0,2,-1,0,-2), (0,0,2,-1,0), (3,3,0,-5,2), (1,1,0,-3,2), (1,1,0,1,-2)} and the dimension of U+V is 6.

To find U ∩ V, we need to find the set of vectors that are in both U and V. From inspection, we can see that the vector (1,1,0,0,0) is in both U and V. Therefore, a base for U ∩ V is {(1,1,0,0,0)} and the dimension of U ∩ V is 1.

No, U and V do not define the same vectorial subspace because they have different bases and dimensions. They also have different generating vectors.