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LAHW#01

LAHW#01. Due October 3, 2011. 1.1 Solving Systems of Linear Equations. 10. Solve the system of equations whose augmented matrix is. 1.1 Solving Systems of Linear Equations. 20. Find the reduced row echelon form of these matrices:. 1.1 Solving Systems of Linear Equations. 26.

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LAHW#01

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  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!

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