Math 308 Midterm 1 Review Sheet
(a) Know what a system of linear equations is. Know the definitions of the augmented matrix and coefficient matrix for a system of linear equations. (1.1: 1-6, 24 - 29)
(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)
4. In particular, know how to multiply matrices and the basic properties of
(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)