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LAHW#01 Due October 3, 2011 1.1 Solving Systems of Linear Equations • 10. – Solve the system of equations whose augmented matrix is 1 2 1 0 3 1 0 3 5 4 3 10 1.1 Solving Systems of Linear Equations • 20. – Find the reduced row echelon form of these matrices: 0 4 a. 0 8 3 c. 4 6 5 12 0 0 3 16 b. 0 1 7 20 3 1 6 12 1 3 4 6 2 d . 1 2 1 1 3 0 0 1 2 3 1 0 3 2 4 3 1 3 1.1 Solving Systems of Linear Equations • 26. – Explain: If a system of linear equations has exclusively rational numbers for the data aij and bi, and if the system has a solution, then it will have a rational solutions. (A real number is said to be rational if it can be expressed as the quotient of two integers.) 1.1 Solving Systems of Linear Equations • 46. – Establish that if a matrix has all integer entries, then it is row equivalent to a matrix in row echelon form having only integer entries. Can we make the same assertion for the reduced row echelon form? 1.2 Vectors and Matrices • 2. – Let A = 3 7 4 5 2 6 2 1 1 1 2 4 and let b be a vector in R4 such that the system Ax = b has a solution. Explain why it has only one. 1.2 Vectors and Matrices • 3. (Continuation.) – Let A be as in General Exercise 2, and let b = [68, –32, 15, 4]T and x = [2, 6, –5]T. The superscript T indicates that these vectors are to be considered as column vectors. Determine whether x is a solution of the system Ax = b. 1.2 Vectors and Matrices • 22. – In this problem, we describe matrices by listing their columns, which are vectors in Rm. Explain a b b b and k < n, why if a a ak b1 b2 bk . If this turns out to then a1 a2 be false, provide a suitable example. 1 2 n 1 2 n 1.2 Vectors and Matrices • 41. – Let A and B be m × n matrices. Explain why A = B if and only if Ax = Bx for all x in Rn. Half of this (the only if) part is rather obvious. It is the if part that requires an idea!