SUNY Oswego Matrix Algebra Flash Cards
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| What are the possible solutions for a system of n equations in m variables? | 3 -
* Unique Solution
* No Solution
* Infinetly Many Solutions
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| What is an example of an unique solution? | ax+by=c
dx+ey=f
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| What is an example of no solution? | ax+by=c
ax+by=d
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| What is an example of Infinetly Many Solutions? | ax+by=c
k(ax+by=c)
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| What is pivoting? | Multiplying a row by a number to get rid of a column.
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| 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.
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| 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.
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| 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.
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| 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
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| R.E.F. | |#****|
|0#***|
|00#**|
|000#*|
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| R.R.E.F. | |100*0|
|010*0|
|001*0|
|00001|
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| Theorem 1 | Any matrix has 1 and only 1 R.R.E.F.
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| Theorem 2 | Any system of linear equations has a unique solution, has infinetly many solutions or else has no solutions.
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| 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.
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| 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
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| 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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