Mathematics
Mathematics, 14.09.2019 07:20, teyante7301

Fix a matrix a and a vector b. suppose that y is any solution of the homogeneous system ax=0 and that z is any solution of the system ax=b. show that y+z is also a solution of the system ax=b.

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Mathematics, 19.07.2019 06:10, ejones123
X(o) = -7 0 0 0 1-3 40 4 0 1 0 -5 0 xo) 4 l 2 1 4 -1 x = points) use theorem i on page 350 to solve the above system of differential equations (see section 5.6 video). 2. (33points) use your solution to show that your solution solves the original system of differential equations. 350 chapter 5 linear systems of differential equations fundamental matrix solutions because the column vector (1) of the fundamental matrix in (2) satisfies the differential equation x = axit follows (from the definition of matrix multi- plication) that the matrix x= (/) itself satisfies the matrix differential equation x = ax. because its column vectors are linearly independent, it also follows that the fundamental matrix () is nonsingular, and therefore has an inverse matrix (0) conversely, any nonsingular matrix solution ) of eq. (1) has linearly independent column vectors that satisfy eq. (1), so () is a fundamental matrix for the system in (1). in terms of the fundamental matrix () in (2), the general solution (3) x(1) = 2x: (0) + 0x2(0)++. (1) of the system x = ax can be written in the form x(i) = (i) (4) sa where cele is an arbitrary constant vector. i is any other fundamental matrix for (1), then each column vector of ) is a linear combination of the column vectors of ), so it follows from eq. (4) that o = c (4) for some x xn matrix c of constants. in order that the solution x(i) in (3) satisfy a given initial condition . k(0) = a, it suffices that the coefficient vector e in (4) be such that (o) = xo: that is, that c= (0)'x when we substitute (6) in eq. (4). we get the conclusion of the following theorem. theorem i fundamental matrix solutions letu) be a fundamental matrix for the homogeneous linear system x = then the (unique solution of the initial value problem x x = axx(o) = x is given by x) = (0) 'xo. section 5.2 tells us how to find a fundamental matrix for the system with constant coefficient matrix a, at least in the case where a has a com- plete set of linearly independent eigenvectors vi. y. associated with the
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Mathematics, 18.09.2019 02:30, jaays4286
Suppose an n×n matrix a has determinant equal to 1. which of the following statements is true? check the correct answer(s) below. a. the system ax=b has a unique solution for all b in rn b. the matrix a is invertible c. the reduced row echelon form of the matrix a is the identity matrix d. the matrix a is singular. e. the homogeneous system ax=0 has infinitely many solutions f. none of the above
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Mathematics, 20.09.2019 23:20, Ree0628
Fix a matrix a and a vector b. suppose that y is any solution of the homogeneous system ax=0 and that z is any solution of the system ax=b. show that y+z is also a solution of the system ax=b.
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Mathematics, 15.10.2019 18:00, CaylaJosephinee
Mark each statement true or false. justify each answer. a. a homogeneous system of equations can be inconsistent. choose the correct answer below. o a. true. a homogeneous equation can be written in the form ax o, where a is an mxn matrix and 0 is the zero vector in r". such a system ax -0 always has at least one solution, namely x-0. thus, a homogeneous system of o b. true. a homogeneous equation cannot be written in the form ax 0, where a is an mxn matrix and o is the zero vector in r. such a system ax 0 does not have the sclution x 0. thus, a homogeneous system of equations can be o c. false. a homogeneous equation cannot be written in the form ax-0, where a is an mxn matrik and o is the zero vector in , such a system ax 0 does not have the solution x 0. thus, a homogeneous system of equations cannot equations can be inconsistent inconsistent. d. fa se. a homogeneous equation can be w en n the form ax-0, where a is an m x n matrix and。เร the zero vector in such a syster b. if x is a nontrivial solution of ax 0, then every entry in x is nonzero. choose the correct answer below. o a. true. a nontrivial solution of ax o is a nonzero vector x that satisfies ax 0. thus, a nontrivial solution x cannot have any zero entries ax: o always has atleast ne solution, namely r- . us, a homogeneous system of equations cannot be inconsistent. b. falso. a nontrivial solution o, ax-ο is a nonzero vector x that satisfies ax 0. thus, a non ri ial solution x can have some zoro en ries so long as not all of its o tries are zero. true. a nontrivial solution o ax-o is a nonzero vector x hat satisfies ax=0. thus, a non rval solution x can have some zero entres so long as not all of its entries are zero. false. a nontrivial solution of ax = 0 is the zero vector. thus, a nontrivial solution x must have all zero entries. o c d. c. the effect of adding p to a vector is to move the vector in a direction parallel to p. choose the correct answer below. oa. false. given v and p in r2 or r3, the effect of adding p to v is to move v in a direction parallel to the plane through p and o o b. false. given v and p in r2 or r3, the effect of adding p to v is to move v in a direction parallel to the line through v and 0 o c. false. given v and p in r2or r3, he effect of adding p to v is to move v in a direction parallel to the plane through v and o d. true. given v and p in r2or r3 the effect of adding p to v is to move v in a direction parallel to the line through p and 10 d. the equation ax- b is homogeneous if the zero vector is a solution. choose the correct answer below. a. false, system o linear equations said to be homogeneous it can be written in the or m ax-o where aisan men matrix and 0 s the zero vector in r the zero ector sa so ution, th n x a 0 nic s alse b true. a s stem o linear equations is said to be homo eneous r t can be written in the orm ax·0, w ere a s ar m n matrix a dos ezero vector in r rthez o vector sa so on, then b- eao o c. false. a system o linear equations is said to be homogeneous t can be written in the orm ax = b, where a is an m x n matrix and b is a nonzero vector i r thus, the zero vector is never a solution o a homogeneous s stem. od. true. a system of linear equations is said to be homogeneous if it can be written in the form a b, where a is an mxn matrix and b is a nonzero vector in rm. if the zero vector is a solution, then b o e. if ax = b is consistent, then the solution set of ax = b is obtained by translating the solution set of ax = 0-choose the correct answer below. o a. true. suppose the equation ax bis consistent for some given b. then the solution set of ax- b is the set of all vectors ofthe form w-pth, where v, is not a solution of the homogeneous equation ax-0. ob. true. suppose the equation ax b is consistent for some given b, and let p be a solution. then the solution set of ax b is the set of all vectors of the form w ptn, where v, is any solution of the homogeneous equation ax c. false. suppose the equation ax b is consistent for some given b. then the solution set of ax -b is the set of all vectors of the form w ptvh, where vh is not a solution of the homogeneous equation axo o d false. suppose the equation a = b s consistent or some given b, and let p be a solution. then the solution set o agz b is he set o an vectors o the orm w--vh where h і any solution o the homogeneous equation ax-0.
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