MATH 217 Study Guide - Midterm Guide: If And Only If, Orthogonality

TRUE-FALSE, Determine whether each statement below is either true of false. Write either TRUE or FALSE (all caps), as appropriate, before each one. a) ____ A linear system with a reduced row echelon form having a row of zeroes has no solutions. b) ____ No set of 6 dependent vectors in R^5 spans R^5. c) ____ Let V be a vector space and S be a subset of V. If V = span(S), then s is a basis for V. d) _____ If u middot v = 0 then either u = 0 or v = 0. e) _____ If v is a 5-component vector of norm 4, then ||3v|| = (3)^5 4 f) _____ The solution set of a consistent nonhomogeneous linear sysem Ax = b, where A is an m times n matrix and b is m times 1, is a subspace of R^n g) ____ A linear system with a reduced row echelon form having a row of zeroes has infinitely many solutions. h) _____ If u, v and w are all 5-component vectors then (u middot v) middot w = u middot (v middot w) h) ____ If u, v and w are all 5-component vectors then (u middot v) middot w = u middot (v middot w) i) ____ The general solution of the nonhomogeneous linear system Ax = b can be obtained by adding b to the general solution of the homogeneous linear system Ax = 0.

Let V be a nonemp an addition operation and a scalar multiplication operation with scalars V a vector space over F, provided the following ten pty set (whose elements are called vectors) on which are defined in F. We call conditions are satisfied: AL. Closure under addition: For each pair of vectors u and v in V, the sum u under addition: For each pair of vectors u and v in V, the sumu+v is also in V. We say that V is closed under addition. 2. Closure under scalar multiplication: For each vector v in V and each scalar k n F, the scalar multiple kv is also in V. We say that V is closed under scalar multiplication A3. Commutativity of addition: For all u, v V, we have u+v=v+u. A4. Associativity of addition: For all u, v, w e V, we have (u + v) + w = u + (v + w). A5. Existence of a zero vector in V: In V there is a vector, denoted 0, satisfying v+0=v, for all v E V. A6. Existence of additive inverses in V: For each vector v In V, there is a vector denoted -V, in V such that v+1-v) = 0. A7. Unit property: For all veV Iv=v. A8. Associativity of scalar multiplication: For all v e V and all scalars r, sEF. (rs)vr(sv). A9. Distributive property of scalar multiplication over vector addition: For all u V E V and all scalars r EF r(u + v) = ru + rv. A10. Distributive property of scalar multiplication over scalar addition: For all v e V and all scalars r,SEF (r+s)v = rv + sv.

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