lecture 5 - linear systems Flashcards
what is a system of linear equations?
multiple equations forming into a complete matrix
What is a complete matrix?
A matrix, symbolised as A^c, that is made of the regular incomplete A matrix as well as the “known” b column.
What is cramers theorem?
if A is non-singular (det != 0), then there is only one solution that can satisfy all linear equations in the system simultaneously. If this is true, it is considered a cramer system.
What is a hybrid matrix?
Symbol = A(i)
Matrix of A with respect to i that substites the ith column for b (known values).
a(1,1) a(1,2) … b(1) .. a(1,n)
What is cramers method?
If matrix is cramer system, then you can find the solutions to the linear system by finding det(hybrid matrix(i)) / det(A).
> this means replacing each column vector with the known values (b) to find determinant values for x, y and z depending on their position.
What is equivalent systems?
2 systems Ax = b and Cx = b that have some solutions which are equivalent.
> transformations may be needed to make matrix become equivalent.
What is gaussian elimination?
Transform complete matrix using 3 properties in which do not change the solution but will give solutions to the linear system by creating a triangular system.
What are the 3 transformations for gaussian elimination?
E1 = swap 2 rows with eachother
E2 = multiplication of row by scalar value
E3 = substitute rows by a sum of row a(i) and another row.
The focus should be on the unknown values (incomplete matrix a) but should change the known values as well.
What is a triangular system?
Remove values sequentially using transformations to get to a(n) on its own.
1 1 1 1 | 2
0 2 3 2 | 3
0 0 1 1 | 7
0 0 0 3 | 5
Why use gaussian elimination instead of cramers method?
Gaussian elimination is easy for a computer to do and is useful on larger matrices with more elements.
How to get solutions from gaussian elimination?
Use the last x value to find each sequential step prior to it with the transformed known values.
What should be the general method for Gaussian Elimination?
- get leading 1 in first column
- make rest of the column 0
- move to next value.
