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Math 308 Midterm 1 Review Sheet

1. Solving systems of linear equations.

(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 equations :
 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.

(c) Know the elementary operations for manipulating systems of equations and the
corresponding row operations for simplifying matrices . (1.1: 30 - 36)

(d) Be able to recognize when a matrix is in echelon form and reduced echelon form.
(1.2: 1 - 10)

(e) Know how to simplify any matrix to reduced echelon form using row operations.
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 page 29
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
many solutions.
  ii. A homogeneous system either has exactly one solution, or infinitely many
solutions.
  iii. If m < n, then an (m n) homogeneous system must have infinitely many
solutions.

(k) For a system of linear equations know how to find conditions on so
that the system is consistent. (1.3: 24)

2. Applications.

(a) Know how to write a system of linear equations for the traffic flows through a
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 matrix
multiplication:

(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 dependent. I
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)

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