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Math 308 Midterm 1 Review Sheet
1. Solving systems of linear equations.
(b) Be familiar with the three possibilities for any system of linear
i. The system is inconsistent (ie. has no solutions).
ii. The system has exactly one solution (ie. has a unique solution).
iii. The system has infinitely many solutions.
(d) Be able to recognize when a matrix is in echelon form and reduced echelon
(1.2: 1 - 10)
(e) Know how to simplify any matrix to reduced echelon form using row
If you have trouble with this, then look at the instructions on page 20 of the
text. (1.2: 22 - 35)
(f) Know how to recognize immediately from the reduced echelon form of the
augmented matrix for a system of equations which of the three possibilities from
(b) applies. Also, you should be able to immediately state how many dependent
and independent variables there are in a system from the reduced echelon form.
(1.3: 1 - 4)
(g) Know how to write the general solution to a system of equations from the
reduced echelon form of the augmented matrix. (1.2: 11 - 21)
(h) Know the definition of the rank of a matrix (defined near the bottom of
in the text, it is r). (1.3: 5, 6)
(i) Know the definition of a homogeneous system of linear equations.
(j) Remember the following facts: (1.3: 7 - 22)
i. If m < n, then an (m*n) system is either inconsistent, or has infinitely
ii. A homogeneous system either has exactly one solution, or infinitely many
iii. If m < n, then an (m n) homogeneous system must have infinitely many
(k) For a system of linear equations
know how to find conditions on
that the system is consistent. (1.3: 24)
(a) Know how to write a system of linear equations for the traffic flows
grid. (1.4: 1 - 4)
(b) Know how to use Kircho 's laws and Ohm 's law to write a system of linear
equation for the currents through an electric circuit. (1.4: 5 - 8)
(c) Know how to write a system of equations to find an interpolation for a given
set of data. (1.8: 1 - 6)
3. Know the basic matrix operations, and their properties:
(a) Matrix multiplication. (1.5: 31 - 41)
(b) Addition of matrices . (1.5: 1 - 6)
(c) Scalar multiplication.
(d) Transpose. (1.6: 7 - 11, 30 - 31)
(a) The associative law : A(BC) = (AB)C.
(b) The distributive laws :
i. A(B + C) = AB + AC
ii. (A + B)C = AC + BC
(c) For any scalar k, k(AB) = (kA)B = A(kB)
(d) Matrix multiplication is not commutative. (ie there are matrices A and B where
AB does not equal BA.)
(e) There exist nonzero matrices A and B so that AB = 0.
5. Be familiar with the identity matrix I and its main property. This
property is: For
any matrix A where the multiplication is defined, AI = A and IA = A.
6. Know how to tell if a set of vectors is linearly independent or linearly
guarantee that you will be asked to do this on the test. (1.7: 1 - 14)
7. Remember the following fact about linear dependence. If
is a set of n
vectors in Rm and m < n, then the set is linearly dependent. Know
why this is true (it's equivalent to the fact that an (m*n) homogeneous system of
equations must have nontrivial solutions).
8. Know the five equivalent conditions that mean an n*n matrix A is nonsingular:
(a) The columns of A are linearly independent.
(b) The only solution of
(c) For any vector in Rn, the system of equations has a unique solution.
(d) The reduced echelon form of A is the identity matrix I.
(e) There exists a unique matrix A-1 such that AA-1 = I.
Be able to use these to determine whether a matrix is nonsingular or singular. (1.9: 9 - 12, 27 - 28)
9. Know how to calculate the inverse of a nonsingular matrix. (1.9: 13 - 21)