309D Test 1 Study Guide
Problem #1:
(a)-[20 pts. ] Solve a system of linear algebraic equations similar to the following example:
x yz 0
2x  y  2z  1
x  2 y  z  1
Hint: Use row operations to bring the augmented matrix of the given system to reduced
row echelon form.
Read Section 1.1 & 1.2.
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Problem #2:
(a)-[15 pts.] Find the general solution of a system of linear algebraic equations similar to
the following:
x1  x2  2 x3  0
x1  x3  x4  0
2 x1  x2  3x3  x4  0
(b)-[5 pts.] Write the general solution of the given system in vector parametric form.
See section 1.5, example 3, p. 52.
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_Problem 3:
 b1 
(a)-[20] Under what condition on the components of the vector b  b2  is the system
 
b3 
 a11 a12 a13 
 x1 


Ax  b consistent? Here A   a21 a22 a23  , and x   x2  . The elements of A will be
 a31 a32 a33 
 x3 
given numbers.
Hint: Bring the augmented matrix of the given system into reduced row echelon form.
See section 1.4, Example 3, p. 43.
309D Test 1 Study Guide
Problem 4:
Write a given linear algebraic system of 3 equations in 3 unknowns similar to the following:
x1  x2  2 x3  b1
x2  x3  b2
 x1  x2  b3
(a)-[4pts.] in vector form
(b)-[4 pts.] in matrix form
(c)-[12 pts.] Show that the columns of the matrix of the given system span R3 , i.e. show
that the given system has a unique solution for every vector b in R3 .
See section 1.4
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Problem #5:
Do a given problem similar to #5, #7 or #9 on page 63.
See “Balancing Chemical Equations” in section 1.6 , p. 59.
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