Linear Algebra Question (Linear independence, subspace vectors span, linear combination)?
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Let v_1 = (1, 2, 2, 1), v_2 = (0, 2, 0, 1), v_3 = (-2, 0, -4, 3). a) Show that these vectors are linearly independent. b) What is the subspace of E^4 that they span, that is, given v = (y_1, y_2, y_3, y_4) how can we tell when v is a linear combination of v_1, v_2, and v_3? Please explain all steps. Thank you!
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Answer:
if a v1 + b v2 + c v3 = 0 { vector } then a .........-2 c = 0 2a..+ 2b .....= 0 2a ........- 4c = 0 a...+ b..+3c = 0...1st and 3rd ---> a = 2c ..2nd and 4th ---> c = 0 , thus 0 = a = b and " det { v1 , v2 , v3 , v } = 0 "
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Other answers
if a v1 + b v2 + c v3 = 0 { vector } then a .........-2 c = 0 2a..+ 2b .....= 0 2a ........- 4c = 0 a...+ b..+3c = 0...1st and 3rd ---> a = 2c ..2nd and 4th ---> c = 0 , thus 0 = a = b and " det { v1 , v2 , v3 , v } = 0 "
ted s
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