(3) (1.2: Solving linear equations) Consider the following linear system with a and b
unknown non-zero constants.
x_(1)-3x_(2)+x_(3)=4
2x_(1)-8x_(3)=-2
-6x_(1)+6x_(2)+ax_(3)=b
(a) For what values of a and b does the system have infinitely many solutions?
(b) Given an example of a and b where the system has exactly one solution.
(c) Give an example of a and b for which the system has no solutions.
(4) (after 2.1) Find a 3 imes 4 matrix A, in reduced echelon form, with free variable x_(3),
such that the general solution of the equation Ax=[[-1],[1],[6]] is
x=[[-1],[1],[0],[6]]+s[[-1],[2],[1],[0]]
where s is any real number.
(5) (after 2.2)
(a) The set
P={[[x_(1)],[x_(2)],[x_(3)]]:2x_(1)-x_(2)+4x_(3)=0}
is a plane in R^(3). Find two vectors u_(1),u_(2)inR^(3) so that span{u_(1),u_(2)}=P.
Explain your answer.
(b) Consider the three vectors u_(1)=[[2],[7],[-1]],u_(2)=[[3],[2],[1]],u_(3)=[[-5],[8],[-5]]. Let b=[[b_(1)],[b_(2)],[b_(3)]]
be an arbitrary vector in R^(3). Use Gaussian elimination to determine which
vectors b are in span{u_(1),u_(2),u_(3)}.
Without further calculation, conclude that span{u_(1),u_(2),u_(3)} is a plane in R^(3)
and identify an equation of the plane in the form ax_(1)+bx_(2)+cx_(3)=0.
(3) (1.2: Solving linear equations) Consider the following linear system with a and b unknown non-zero constants. X1 3x2 + x3 = 4 2x1 8x3 2 -6xi + 6x2 + ax3 = b
(a) For what values of a and b does the system have infinitely many solutions? (b) Given an example of a and b where the system has exactly one solution. (c) Give an example of a and b for which the system has no solutions.
(4) (after 2.1) Find a 3 4 matrix A, in reduced echelon form, with free variable x3, -1 such that the general solution of the equation Ax = 1 is 6
1
2 1
X =
+s
0 6
where s is any real number. (5) (after 2.2) (a) The set
X2 : 2x1 - x2 + 4x X3
is a plane in R3. Find two vectors ui, u2 E R3 so that span{u1,u2} = P Explain your answer.
2
[3] 2 u3=
[b1] (b) Consider the three vectors u1 = u2 8 . Let b = b2 b3] be an arbitrary vector in R3. Use Gaussian elimination to determine which vectors b are in span{u1,u2,u3}. Without further calculation, conclude that span{u1,u2, u3} is a plane in R3 and identify an equation of the plane in the form ax1 + bx2 + ct3 = 0.
