# Dimension of U,V and W over K: Do they Equal?

• garyljc
In summary, the dimension of U, V, and W over K refers to the number of linearly independent vectors needed to span the vector space over the field K. This can be determined by finding the maximum number of linearly independent vectors in each vector space. The dimensions of U, V, and W can be different, and the dimension of U+V is equal to the sum of the dimensions of U and V minus the dimension of their intersection. The dimension of U, V, and W over K is significant in determining the basis and span of the vector space, as well as describing the number of independent variables needed in linear algebra.
garyljc
Just have a question
if U,V and W are over the same field K
does it mean that dim U = dim V = dim W ?

No.

For example, $R^2$ and $R^3$ are both over the same field R, but R^2 has dimension 2 and R^3 has dimension 3.

## 1. What is the definition of "Dimension of U,V and W over K"?

The dimension of U, V, and W over K refers to the number of linearly independent vectors needed to span the vector space U, V, and W, respectively, over the field K.

## 2. How do you determine the dimension of U, V, and W over K?

The dimension of U, V, and W over K can be determined by finding the maximum number of linearly independent vectors in each vector space. This can be done by row reducing the matrix formed by the vectors and counting the number of non-zero rows.

## 3. Can the dimensions of U, V, and W be different?

Yes, the dimensions of U, V, and W can be different. Each vector space can have a different number of linearly independent vectors, therefore resulting in different dimensions.

## 4. How does the dimension of U+V compare to the dimensions of U and V separately?

The dimension of U+V is equal to the sum of the dimensions of U and V minus the dimension of their intersection. This can also be written as dim(U+V) = dim(U) + dim(V) - dim(U∩V).

## 5. What is the significance of the dimension of U, V, and W over K?

The dimension of U, V, and W over K is important because it tells us how many independent variables are needed to describe the vector space. It also helps determine the basis and span of the vector space, which are essential concepts in linear algebra.

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