1.5 - 1.5 Solution Sets of Linear Systems Before proceeding...

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Unformatted text preview: 1.5 Solution Sets of Linear Systems Before proceeding, do a quick Review: For vectors in R 3 , (i) Span { vector v 1 , vector v 2 , vector v 3 } = { c 1 vector v 1 + c 2 vector v 2 + c 3 vector v 3 } is R 3 in many (most) cases. (ii) Span { vector v 1 , vector v 2 } = { c 1 vector v 1 + c 2 vector v 2 } is a plane in R 3 in many (most) cases. (iii) Span { vector v 1 } = { c 1 vector v 1 } is a line R 3 in many (almost all) cases — viz., unless vector v 1 = vector 0. Q: Is the vector vector 0 in Span { vector v 1 , vector v 2 } or Span { vector v 1 } ? A: Yes, the weights can be zero. So pictorially, the plane or line above always goes through the origin. EX: From the last example in the previous section, Span 1 , 1 1 = a 1 + b 1 1 = a b b for every a , b ∈ R It goes through the origin. · For x = 0, it is the line y = z 1 · For any x = a , it is the line through ( a, , 0) parallel to y = z . x y z Now back to section 1.5: homogeneous linear system Avectorx = vector , i.e. , vector b = vector Q: Is a homogeneous linear system consistent?...
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1.5 - 1.5 Solution Sets of Linear Systems Before proceeding...

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