Which set could you use Prove that if S vvector1 vvector2 e

Which set could you use? Prove that if S = {v^vector_1, v^vector_2, ellipsis, v^vector_n}is a linearly independent set from R^m then v^vector_n is not a member of span {v^vector_1, v^vector_2, ellipsis, v^vector_n-1} Complete the last part of The Equality of Spans Theorem: Suppose that every w^rightarrow_j can also be written as a linear combination of the v^vector _1 through v^vector_n Show that every member of Span({W^rightarrow_1, W^rightarrow_2, ellipsis, W^rightarrow_m}) is likewise a member of Span ({v^rightarrow_1, v^rightarrow_2, ellipsis, v^rightarrow_v}). Imitate the proof of the first part found in the text, but be careful not to confuse m and n

Solution

50. Let us assume that v_n belongs to span {v₁,v₂,…,v_n-1}. Then v_n can be expressed as a linear combination of v₁,v₂,…,v_n-1. Let v = a₁ v₁+a₂v₂+…+a_n-1 v_n-1, where a₁,a₂,…,a_n-1 are scalars, not all zero. Then, v - a₁ v₁ -a₂v₂ -…-a_n-1 v_n-1 = 0. This means that the vectors v₁,v₂,…,v_n-1,v_n are linearly dependent. This is a contradiction as S = {v₁,v₂,…,v_n-1,v_n} is a linearly independent set. Hence v_n cannot belong to span {v₁,v₂,…,v_n-1}.

51. If every w_j can be expressed as a linear combination of v₁,v₂,…,v_n, then by the definition of a spanning set, every w_j span{ v₁,v₂,…,v_n }. Now, let w = a₁w₂ + a₂w₂+…+a_mw_mbe an arbitrary vector in span{ w₁,w₂,…,w_m}, where a₁,a₂,…,a_m are scalars, not all zero. Since each w_j , 1 j m, is a linear combination of v₁,v₂,…,v_n, hence w = a₁w₂ + a₂w₂+…+a_mw_m is also a linear combination of v₁,v₂,…,v_n. Since w is an arbitrary vector in span{ w₁,w₂,…,w_m}, this implies that every member of span { w₁,w₂,…,w_m} is a member of span{ v₁,v₂,…,v_n }.