# Showing that two vector spaces are equal.

• jdinatale
In summary, two vector spaces are equal if they have the same dimension and contain the same set of vectors. To show that two vector spaces are equal, you must prove that they have the same dimension and that every vector in one space can be written as a linear combination of vectors in the other space. Two vector spaces can still be equal even if they have different bases, as long as they span the same space. Proving that two vector spaces are equal is important because it establishes a sense of equivalence between the two spaces, making it easier to solve problems and make connections between different concepts. However, two vector spaces cannot be equal if they have different dimensions, as the dimension represents the number of linearly independent vectors in a space.
jdinatale
I thought that I should do containment in both directions. I have containment in one direction, but the other is much harder. Any ideas?

Call ##w_1= v_1,\ w_2 = c_{12}v_1+v_2,\ w_3 = c_{13}v_1+c_{23}v_2 + v_3## Can you solve for the ##v_i## in terms of the ##w_i##?

## 1. What does it mean for two vector spaces to be equal?

Two vector spaces are equal if they have the same dimension and contain the same set of vectors. This means that they have the same number of linearly independent vectors and span the same space.

## 2. How do you show that two vector spaces are equal?

To show that two vector spaces are equal, you must prove that they have the same dimension and that every vector in one space can be written as a linear combination of vectors in the other space. This can be done by showing that the basis vectors of one space can be expressed as a linear combination of the basis vectors of the other space.

## 3. Can two vector spaces with different bases be equal?

Yes, two vector spaces with different bases can still be equal as long as they span the same space. This means that the basis vectors of one space can still be expressed as a linear combination of the basis vectors of the other space, even though the basis vectors themselves may be different.

## 4. What is the importance of proving that two vector spaces are equal?

Proving that two vector spaces are equal is important because it allows us to establish a sense of equivalence between the two spaces. This means that any theorems or properties that hold for one space will also hold for the other space, making it easier to solve problems and make connections between different concepts.

## 5. Can two vector spaces be equal if they have different dimensions?

No, two vector spaces cannot be equal if they have different dimensions. This is because the dimension of a vector space represents the number of linearly independent vectors it contains. If the two spaces have different dimensions, then they cannot contain the same set of vectors and therefore cannot be equal.

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