LAHW#01

1. LAHW#01 Due October 3, 2011

2. 1.1 Solving Systems of Linear Equations • 10. • Solve the system of equations whose augmented matrix is

3. 1.1 Solving Systems of Linear Equations • 20. • Find the reduced row echelon form of these matrices:

4. 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.)

5. 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?

6. 1.2 Vectors and Matrices • 2. • Let A = and let b be a vector in R4 such that the system Ax = b has a solution. Explain why it has only one.

7. 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.

8. 1.2 Vectors and Matrices • 22. • In this problem, we describe matrices by listing their columns, which are vectors in Rm. Explain why if and k < n, then . If this turns out to be false, provide a suitable example.

9. 1.2 Vectors and Matrices • 41. • Let A and B be m × n matrices. Explain why A = Bif and only ifAx = 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!