1. Determine whether or not the following linear system of 4 equations in 5 variables x1,x2,x3,x4,x5 has a solution

Task #1 1. Decide whether or not or not the next linear system of four equations in 5 variables x1,x2,x3,x4,x5 has an answer. x1 + 2x2 + 3x3 + 4x4 + 5x5 = 6 -x1 + x2 - x3 + x4 - x5 = 1 x2 + x4 = 2 x1 + x3 + x5 = 1 If there may be precisely one answer, clear up for it. If there are infinitely many options, parameterize the answer set. If there aren't any options, say that there aren't any options. 2. [Take into account the next linear system of two equations in four variables x , x , x , x . x1 + 2x3 - 4x4 = zero x2 + 3x4 = zero The answer set of this linear system might be parametrized utilizing two parameters t1,t2 and two vectors v1,v2 within the kind s(t1,t2) = t1v1 + t2v2. Discover the vectors v1,v2 given the knowledge that and . three. Denote by v1, v2, v3, and v4 the next vectors of R3: , and . (a) Is the span Spanv , v , v , v all of R3? Why or why not? (b) Decide if the vectors v , v , v , and v kind a linearly dependent or linearly impartial set of vectors. Justify your reply. four. Present counterexamples to the next claims. (a) Each linear system of three equations in three variables has a singular answer. (b) There don't exist three vectors in R3 which are linearly impartial. (c) There aren't any linear transformations T : R3 ?R4 which are one-to-one. (d) If a linear transformation T : R3 ?R2, is onto then T is one-to-one. 5. Let T : R4 ?R3 be the linear transformation outlined by the components T(x) = Mx the place the matrix M is outlined as and x is any vector of R4. (a) Present that T is onto. (b) Present that T isn't one-to-one. (c) Give a parametrization for the answer set of the linear system outlined by the equation Mx = v the place ?x1? x = ??xx23??? and. ? x4 2
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