Linear Algebra - (Direct Sum | Union) of vector spaces

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Let U and V be two vector spaces consisting of D-vectors over a field F.

Definition: If U and V share only the zero vector then we define the direct sum of U and V to be the set: <MATH> \{u + v : u \in U, v \in V\} </MATH> written: <MATH> U \oplus V </MATH> That is, <math>U \oplus V</math> is the set of all sums of a vector in U and a vector in V.


In Python, [u+v for u in U for v in V]


  • <math>U \oplus V</math> is a vector space
  • The union of a set of generators of U, and a set of generators of V is a set of generators for <math>U \oplus V</math>
  • Union of a basis of U and a basis of V is a basis of <math>U \oplus V</math> because the union is linearly independent



<math>dim U + dimV = dimU \oplus V</math>


over Gf2

Vectors over GF(2):

Let U = Span {1000, 0100} and let V = Span {0010}.

  • Every non-zero vector in U has a one in the first or second position (or both) and nowhere else.
  • Every non-zero vector in V has a one in the third position and nowhere else. Therefore the only vector in both U and V is the zero vector.

Therefore <math>U \oplus V</math> is defined.

<MATH> \begin{eqnarray*} U \oplus V & = \{& 0000+0000,& 1000+0000, &0100+0000, &1100+0000, &0000+0010, &1000+0010, &0100+0010, &1100+0010\} \\ & = \{&0000,& 1000,& 0100, &1100, &0010, &1010, &0110, &1110\} \end{eqnarray*} </MATH>

over <math>\mathbb{R}</math>

  • Let U = Span {[4,−1, 1]}.
  • Let V = Span {[0, 1, 1]}.

The only intersection is at the origin, so <math>U \oplus V</math> is defined.

<math>U \oplus V</math> is the set of vectors u + v where <math>u \in U \text{ and } v \in V</math>

This is just Span {[4,−1, 1], [0, 1, 1]}, Plane containing the two lines

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