Abstract Algebra: Theory and Applications · Problem SE.20.1

Fields — Thomas Judson

Mathematics — Abstract Algebra · subspaces-dimension

Problem

Given two subspaces

U

and

W

of a vector space

V

, their sum

U + W

can be defined as the set

U + W = \{u + w \mid u \in U, w \in W\}

, in other words, the set of all possible sums of an element from

U

and an element from

W

. Notice this is not the direct sum of your text, nor the direct\_sum() method in Sage. However, you can build this subspace in Sage as follows. Grab the bases of

U

and

W

individually, as lists of vectors. Join the two lists together by just using a plus sign between them. Now build the sum subspace by creating a subspace of

V

spanned by this set, by using the .subspace() method. In the vector space \texttt{(QQ\^10)} construct two subspaces that you expect to (a) have dimension 5 or 6 or so, and (b) have an intersection that is a vector space of dimension 2 or so. Compare their individual dimensions with the dimensions of the intersection of

U

and

W

(

U \cap W

, .intersection() in Sage) and the sum

U + W

. Repeat the experiment with the two original vector spaces having dimension 8 or so, and with the intersection as small as possible. Form a general conjecture relating these four dimensions based on the results of your two (or more) experiments.

Intuitive explanation

The dimension formula for subspaces—called the rank-nullity theorem's cousin—connects how two subspaces overlap and combine. When you add two subspaces together, their dimensions don't simply add: you must account for their overlap, since vectors in the intersection get "double-counted." By constructing concrete examples and measuring all four dimensions ( $\dim U$ , $\dim W$ , $\dim(U \cap W)$ , $\dim(U + W)$ ), you'll discover a beautiful additive relationship that reveals how intersection and union interact algebraically.

Intuitive explanation

Related theory

Dimension Formulas For Subspaces

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