Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all...
Suppose a nonhomogeneous system Ax = b consisting of five linear equations with seven unknowns has a solution with three free variables. Is it possible to change some constants on the equations' right sides to make the new system inconsistent? Explain.
Is it possible that all solutions of a homogeneous system of twelve linear equations in fifteen variables are multiples of one fixed nonzero solution? Discuss. Consider the system as Ax = 0, where A is a 12 x 15 matrix. Choose the correct answer below. O A. No. Since A has 12 pivot positions, rank A = 12. By the Rank Theorem, dim Nul A = 12-rank A = 0. Since Nul A = 0, it is impossible to find...
o. (5 points) Let AX0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution. Show that if k is any positive integer, then the system A*X0 also has only the trivial solution.
U 10 I LI IU I LILIUI IUL 2. (Bases for column and row spaces) Suppose A is a 3 x 3 matrix with pivots in exactly the first and second columns. Suppose B is the unique reduced echelon form of A. (a) We know that the first and second columns of A form a basis for Col A. Is it necessarily true that the first and second columns of B form a basis for Col A? If so, explain...
66. Suppose a non-homogeneous system AF = 5 of six linear equations in eight variables has a solution, with two free variablea. Is is possible that Až = is inconsistent for some y in R6? Why or why not? 67. Let A be a 4 x 4 matrix. The eigenvectors of A are 6 and - 5. The eigenspace corresponding to 1 = 6 is 2-dimensional and the eigenspace corresponding to A = -5 is 1-dimensional. Is A diagonalizable? Why...
A linear system with fewer equations than unknowns is sometimes called an underde- termined system. Prove that if an underdetermined system has a solution, then it has infinitely many solutions. 7. A linear system with fewer equations than unknowns is sometimes called an underde- termined system. Prove that if an underdetermined system has a infinitely many solutions solution, then it has
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...
Consider the homogeneous system of linear equations 20 3r1 +2r2+40 Verify that the set of solutions is a linear subspace of R3. Find a basis of this subspace.
2019 Summer I 270 Exam 1A Take Home Due at 1120 on 20190520 Write down a system of six distinct linear equations in the unknowns x1, X2、Xy, and x4, in R4 such that: 0 exactly two of the unknowns 1 x2, 3, and x4 occur in each of them with non-zero real coefficients 1 the system has exactly one solution: χ,-c, and x,-d respectively. a, X2-b, Xy Prove that your answer is correct; otherwise, you will get 0 on this...
8. Solve the following first order homogeneous linear system of differential equations I a -B -B Cu = -3 4 -3 | u, 1-B –B a ) where a and B are real nonzero constants. Find a fundamental matrix and the inverse matrix of the fundamental matrix. Hint: dot1 TI 212/1 a 22).