Subspace of 20211004 matrix

Theorem $1.4.$ If $V_1,V_2$ It's a number field $K$ Linear spaces over $V$ Two subspaces of , So their intersection $V_1\cap V_2$ It's also $V$ The subspace of ．

Theorem $1.5.$ If $V_1,V_2$ It's all number fields $K$ Linear spaces over $V$ The subspace of , So their sum $V_1+V_2$ It's also $V$ The subspace of .

Theorem $1.6.$ （ Dimension formula ） If $V_1,V_2$ It's a number field $K$ Linear spaces over $V$ Two subspaces of , Then there is the following formula
$$\dim V_1+\dim V_2=\dim \left(V_1+V_2\right)+\dim \left(V_1\cap V_2\right)$$

Theorem 1.4 A good understanding , For theorem 1.5, First of all, make it clear $V$ yes Linear space , give an example : $V_1$ yes xoy Plane , $V_2$ yes oz A straight line , that $V_1+V_2$ It's not simple xoy Plane +oz A straight line , Because this does not satisfy the linear condition , We can work out $V_1+V_2$ yes o-xyz three-dimensional space .

in other words $V_1+V_2$ Is to find the smallest , contain $V_1$ and $V_2$ Linear space .