Question | Answer |
What are the possible solutions for a system of n equations in m variables? | 3 -
* Unique Solution
* No Solution
* Infinetly Many Solutions |
What is an example of an unique solution? | ax+by=c
dx+ey=f |
What is an example of no solution? | ax+by=c
ax+by=d |
What is an example of Infinetly Many Solutions? | ax+by=c
k(ax+by=c) |
What is pivoting? | Multiplying a row by a number to get rid of a column. |
Matrix Notation | A matrix is a rectangular array of Real numbers represented as a "nxm matrix" which consists of n rows, each of which contains m entries. Likewise, it consists of m columns each of which has n entries. |
What are the three properties of echelon form? | 1) All nonzero rows are above any rows of all zeros.
2) Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3) All entries in a column below a leading entry are zeros. |
What are the conditions of a rectangular matrix in Reduced Row Echelon Form (RREF) | It must meet the 3 properties of echelon form as well as:
4) the leading entry in each nonzero row is 1.
5) Each leading 1 is the only nonzero entry in its column. |
Are row operations reversible? If yes how, if no, why not? | Yes
(i) ri<->rj is "reversed" by rj<->ri
(ii) kri->ri is "reversed" by 1/kri->ri (k=0 is not allowed)
(iii) kri+rj->rj is "reversed" by -kri+rj-)rj |
R.E.F. | |#****|
|0#***|
|00#**|
|000#*| |
R.R.E.F. | |100*0|
|010*0|
|001*0|
|00001| |
Theorem 1 | Any matrix has 1 and only 1 R.R.E.F. |
Theorem 2 | Any system of linear equations has a unique solution, has infinetly many solutions or else has no solutions. |
Correspondence Theorem | The vector b is in the span {a1, a2,...,an} precisely when the linear system with augmented matrix (a1:a2:...:an|b) has a solution.
ie: b=&1a1+&2a2+...+&nan precisely when x1=&1, x2=&2,...,xn=&n is a solution to the linear system. |
What is the matrix equation:
|1 2| |x| |7|
|3 4|*|y|=|8|
|5 6| |9| | x+2y=7
3x+4y=8
5x+6y=9 |
When is column vector defined? | The matrix product Ax where x is an nx1 matrix, column vector is defined only when A is mxn |