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

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The dimensions of vector spaces U, V, and W over the same field K do not necessarily equal each other. An example provided is R^2 and R^3, which are both defined over the field R, yet have dimensions of 2 and 3, respectively. This illustrates that having the same field does not imply equal dimensions among vector spaces. Therefore, dim U can differ from dim V and dim W even when they are over the same field. The concept emphasizes the independence of dimensionality from the underlying field.
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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 ?
 
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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.
 

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