91影视

Recall that a vector space V is the direct sum of two subspaces W鈧� and W鈧�, denoted by V = W鈧� 鈯� W鈧�, if both of the following conditions are satisfied: 1. \(W_{1} \cap W_{2} = \{0\}\), that is, the intersection of the spaces contains only the zero vector. 2. \(W_{1} + W_{2} = V\), that is, every vector in V can be written as the sum of vectors in W鈧� and W鈧�.

Let's assume V is the direct sum of W鈧� and W鈧�, that is, V = W鈧� 鈯� W鈧�. By definition, this means that every vector in V can be written as the sum of vectors in W鈧� and W鈧�. Let v 鈭� V, then there exist unique vectors x鈧� 鈭� W鈧� and x鈧� 鈭� W鈧� such that v=x鈧�+x鈧�. Suppose there is a second representation \(v= x_{1}' + x_{2}'\) with \(x_{1}' \in W_{1}\) and \(x_{2}' \in W_{2}\). Now, we have \(x_{1} + x_{2} = v = x_{1}' + x_{2}'\), which can be rewritten as \(x_{1} - x_{1}' = x_{2}' - x_{2}\). Notice that both \(x_{1} - x_{1}'\) and \(x_{2}' - x_{2}\) belong to the intersection \(W_{1} \cap W_{2}\). Since V = W鈧� 鈯� W鈧�, the intersection \(W_{1} \cap W_{2}\) is only the zero vector. Therefore, we have \(x_{1} - x_{1}' = x_{2}' - x_{2} = 0\), which implies that \(x_{1} = x_{1}'\) and \(x_{2} = x_{2}'\). Therefore, each vector in V can be uniquely written as x鈧�+x鈧� when V = W鈧� 鈯� W鈧�.

Now let's assume that each vector in V can be uniquely written as x鈧�+x鈧�, where x鈧� 鈭� W鈧� and x鈧� 鈭� W鈧�. We need to show that V = W鈧� 鈯� W鈧�. In this case, we know that \(W_{1} + W_{2} = V\) since every vector in V can be written as the sum of vectors in W鈧� and W鈧�. We also need to show that \(W_{1} \cap W_{2} = \{0\}\). Let w 鈭� \(W_{1} \cap W_{2}\), then w can be uniquely written as x鈧�+x鈧� with x鈧� 鈭� W鈧� and x鈧� 鈭� W鈧�. But w is also in W鈧�, hence w=x鈧�+x鈧�=0+x鈧�=x鈧�. Since w 鈭� W鈧�, we have W鈧� 鈭� W鈧� containing zero vector. If there is any other non-zero vector w also belongs to \(W_{1} \cap W_{2}\), then it would contradict the assumption that each vector in V can be uniquely written as x鈧�+x鈧�. Therefore, \(W_{1} \cap W_{2} = \{0\}\), and V = W鈧� 鈯� W鈧�. In conclusion, V is the direct sum of W鈧� and W鈧� if and only if each vector in V can be uniquely written as x鈧�+x鈧�, where x鈧� 鈭� W鈧� and x鈧� 鈭� W鈧�.