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Matrix Algebra
SUNY Oswego Matrix Algebra Flash Cards
| 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 |
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