Download LAHW01

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!