MATH136 Lecture Notes - Lecture 4: Hyperplane, Linear Combination

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ultramarineelephant806

26 Jul 2018

School

University of Waterloo

Department

Mathematics

Course

MATH136

Professor

Alejandra Vicente Colmenares

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Math 136 lecture 4 10 jan, 2018. Def: the standard basis for is denoted {(cid:2869) , , }(cid:1488) where is the vector in where the -th component is 1 and all other entries are 0. Let be a basis for a subset of . Then every (cid:1876) (cid:1488) can be written as a unique linear combination of the vectors in . Definition: let , (cid:1488) , such that { } is linearly independent (means that cannot be (cid:882) ). We define a line in as a set with vector equation (cid:1876) = +(cid:1872) ,(cid:1872)(cid:1488) . Definition: let ,(cid:2869) ,(cid:2870) (cid:1488) , such that {(cid:2869) ,(cid:2870) } is linearly independent. The set (cid:1876) = +(cid:1872)(cid:2869)(cid:2869) +(cid:1872)(cid:2870)(cid:2870) ,(cid:1872)(cid:2869) ,(cid:1872)(cid:2870)(cid:1488) is called a plane in . ,(cid:1872)(cid:2869), ,(cid:1872) (cid:2869)(cid:1488) is called a hyper-plane in . Definition: let ,(cid:2869) (cid:1488) , such that {(cid:2869) , } is linearly independent. The set (cid:1876) = +(cid:1872)(cid:2869)(cid:2869) + +(cid:1872) ,(cid:1872)(cid:2869), ,(cid:1872)(cid:1488) is called a -flat in .

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Let r in and u =(1 1 0), v = (1 4 1), w = (r2 1 r2), b = (3+2r 5+12r 2r). For which values r in is the set {u. v, w} linearly independent? For which values r in is the vector b a linear combination of u. v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set 3 of all column vectors of length three, with real entries, is a vector space. Is the subset B = {(x y z) in 3 | xy + yz = 0} a subspace of 3? Justify your answer. Show that the set of all twice differentiable functions f : satisfying the differential equation sin(x) f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space 3 of all real polynomials p of degree at most 3: S = {p in 3 | p(l) = 0, p'(l) = 0}, where p' is the derivative of p. Determine whether S is a subspace of 3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S. u = (1 1 0), v = (1 4 1), w = (r2 1 r2) if {u, v, w} is linearly independent set 1(4r2 - 1)-1(r2 - r2) ne 0 4r2 ne 1 r ne plusmn1/2 this set is linearly independent set for values of e ne plusmn1/2 - - - - - - - (3+2r 5+12r 2r) = a1 (1 1 0) + a2 (1 4 1) + a3 (r2 1 r2) this forms system of equations in a1, a2, a3 if 4r2 - 1 ne 0 then system has unique solution if 4r2 - 1 = 0 and -2 - 4r = 0 rArr r = -1/2, the system has more than one solution if 4r2 - 1 ne 0 b can be written as linear combination of u,v,w as uniquely if r = -1/2 then b can be written as linear combination of u,v,w more than one way - - - - - - - - - (1 1 -1) in B and (1 0 1) in B (1 1 -1) + (1 0 1) = (2 1 0) not in B since 2times1+1times0 ne 0 so addition of vector is failed, this set is not forms subspace consider the following differential equation (sinx)f"(x) + x2f (x) = 0 each solution is continuous function the set of all solutions is subset of space of set of all continuous functions n from R to R let f,g are two solutions (sinx)f"(x) + x2f (x) = 0 (sinx)g"(x)+x2g(x) = 0 ccheck whether af+bg is solution or not (sinx)f"(x) + x2f(x) = 0 rArr (sinx) af"(x) + x2?f (x) = 0 (sinx)g"(x) + x2g (x) = 0 rArr (sinx)bg" (x) + x2bg (x) = 0 (sinx)(af"(x)+bg"(x)) +x2 (af (x) + bg(x)) = 0 af+bg is also solution to differential equation the set of all solutions to differential equation forms subspace - - - - - - - - - - consider the following subset of P3 {p in P3/.p(l)=0, p'(l) = 0} let p, q in S rArr p(l)= 0, p'(l) = 0 and q(l) = 0, q'(l) = 0 (ap + bq) (l) = ap (l) +b q(l) = a (0) + b (0) = 0 (ap + bq)'(1) = ap'(l) +bq'(1) = a (0) + b(0) = 0 p, q in S rArr ap +bq in S then S is subspace q(x) = x - 2x2 + x3 q(1) = 1 - 2 + 1 = 0 q'(x)=1 - 4x + 3x2 q'(l) = 1 - 4 + 3 = 0 q is element of S

MATH136 Lecture Notes - Lecture 4: Hyperplane, Linear Combination

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