[[L4-text]] V1, V2 Subspaces $\subset$ V V1 $\cap$ V2 is a subspace V1 $\cup$ V2 is usually not a subspace - example: Union of two linear subspaces, the x axis and the y axis; the union is not preserved under the operation of addition. - make diagram - Correct notion is a sum of subspaces - Sum: subset of V : all vectors of the form v1 + v2 all possible combinations, all possible sums v1 $\in$ V1 Sum replaces the notion of Union in sets; Here, you are generating things with addition, and scalar multiplication; Redundancy; add more vectors; add V3 which is sum of v1, v2 Direct Sum defined to address redundancy V1 $\oplus$ V2 $\oplus$ ... $\oplus$Vm if each element can be written uniquely as combination of elements of the subspaces. by v1 + v2 + ... + vm, each in its subspace $\oplus$ Span Linear Dependence Linear Independence 1-m linearly independent u $\in$ V 1- n spanning w $\in$ V Process moving u to W; express u as $b_i$ $w_i$ eliminate one w from spanning set; spanning set still spans V get U set is $\le$ in size to W set and W set is $\ge$ than U set so U set and W set are the same size Process argument