Brian.ning
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- 2013-09-01
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1. Let A be the 9 15 coefficient matrix of a homogeneous linear system, and suppose that
this system has infi nitely many solutions with 8 parameters.
What is the rank of A?
Do the columns of A, considered as vectors in R9, span R9?
2. Find a basis for the solution space of the equation 2x-5y+3z=0
3. If the coefficient matrix A in a homogeneous system of 33 equations in 28 unknowns is known to have rank 12, how many parameters are in the general solution?
4. For a no homogeneous system of 2013 equations in 3012 unknowns, answer the following three questions:
Can the system be inconsistent?
Can the system have infinitely many solutions?
Can the system have a unique solution?
5. Which of two the following three sets is a basis of R3
B1={(1, 0, 1), (6, 4, 5), (-4, -4, 7)}
B2={(2, 1, 3), (3, 1, -3), (1, 1, 9)}
B3={(3, -1, 2), (5, 1, 1), (1, 1, 1)}
6. Suppose e,f in R, and consider the linear system in x, y and z:
3x - 2y + ez = f
x + z = f
2x + y + z = -1
a) if [A|b] is the augmented matrix of the system above, find rank A and rank [A|b] for all values of e and f.
b) using part a) to find all values of e and f so that this system has
i) a unique solution,
ii) infinitely many solutions, or
iii) no solutions.
7. If A= [a b]
[c d]
is a 2*2 matrix such that the vectors [a] and
[c] [d]
are linear independent, prove carefully that rank A=2. (You cannot choose the matrix A - your proof must work for every 2*2 matrix with the property above, i.e. every 2*2 matrix with independent columns.)
this system has infi nitely many solutions with 8 parameters.
What is the rank of A?
Do the columns of A, considered as vectors in R9, span R9?
2. Find a basis for the solution space of the equation 2x-5y+3z=0
3. If the coefficient matrix A in a homogeneous system of 33 equations in 28 unknowns is known to have rank 12, how many parameters are in the general solution?
4. For a no homogeneous system of 2013 equations in 3012 unknowns, answer the following three questions:
Can the system be inconsistent?
Can the system have infinitely many solutions?
Can the system have a unique solution?
5. Which of two the following three sets is a basis of R3
B1={(1, 0, 1), (6, 4, 5), (-4, -4, 7)}
B2={(2, 1, 3), (3, 1, -3), (1, 1, 9)}
B3={(3, -1, 2), (5, 1, 1), (1, 1, 1)}
6. Suppose e,f in R, and consider the linear system in x, y and z:
3x - 2y + ez = f
x + z = f
2x + y + z = -1
a) if [A|b] is the augmented matrix of the system above, find rank A and rank [A|b] for all values of e and f.
b) using part a) to find all values of e and f so that this system has
i) a unique solution,
ii) infinitely many solutions, or
iii) no solutions.
7. If A= [a b]
[c d]
is a 2*2 matrix such that the vectors [a] and
[c] [d]
are linear independent, prove carefully that rank A=2. (You cannot choose the matrix A - your proof must work for every 2*2 matrix with the property above, i.e. every 2*2 matrix with independent columns.)
