Let {v1 rightarrow,...,vn rightarrow} V be a linearly independent set in a vecto
ID: 2941357 • Letter: L
Question
Let {v1 rightarrow,...,vn rightarrow} V be a linearly independent set in a vector space V. Consider another vector W rightarrow that is not in the span of {v1 rightarrow,...,vn rightarrow} Fill in the following outline in order to prove that the larger set {v1 rightarrow ...,vn rightarrow,w rightarrow} is still linearly independent. Say the numbers c1,..., cn, d satisfy the linear equation (1) c1v1 rightarrow + c2v2 rightarrow +. . . .+ cnvn rightarrow + dw rightarrow = 0 What do we need to prove about c1,...,cn and d in order to Conclude that {v1 rightarrow...,vn rightarrow, w rightarrow} is linearly independent? Say d 0 in equation (1). Solve this equation for w. Is it possible for such an equation to exist? What does this tell you about d? If d = 0 in equation (1), what can you say about the other coefficients c1...,cn? EXPLAIN! Bring together your conclusions from the previous steps to Explain why {v1 rightarrow..., vn rightarrow, w rightarrow} is still linearly independent.Explanation / Answer
if d = 0, the other coefficients c1, c2,...,cn must sum to zero for (1) to be true.
if d = 0
c1v1 + c2v2 +...+ cnvn + 0w = 0
c1v1 + c2v2 +...+ cnvn = 0
v1,v2,...vn cannot equal zero, or the solution is trivial.
c1+c2+...+cn = 0
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