3 - week 3 Flashcards
Elementary matrices
An elementary matrix, E, is an n x n matrix obtained by doing exactly one row operation on the n x n identity matrix, I.
Row equivalence
In linear algebra, two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations. Alternatively, two m × n matrices are row equivalent if and only if they have the same row space
Rank of a matrix
(rank can also be used to determing number of dimensions)
Rank(A) is the number of leading ones in a matrix in row-echelon form that is row-equivalent with A.
For an r x c matrix:
If r is less than c, then the maximum rank of the matrix is r.
If r is greater than c, then the maximum rank of the matrix is c.
The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.
Put matrix in REF and then count the 1’s to find the rank
Write a Matrix as a Product of Elementary Matrices
Free variables in vector form
Elementary matrix (must be a square matrix)
Perform 1 row operation from the identy matrix to see if it is a elemtary matrix
Determinant of a matrix
2x2
First diagonal * - second diagonal *
Null space
Brug t variablen vectoren
Remember the row operations
1.
swap 2 rows
2.
multiply a row by a non zero constant OR multiply and add it to another row
3.
Add one row to another
Solving fractions
1 - 1/3
= 2/3
Because 1 whole makes it 3-1 = 2/3
General solution
Determinant of a 3x3 matrix
1.
+ - + checkerboard
2.
take the numbers on the first row and find their submatrix
3.
submatrix is the numbers that are not in the same row and collumn
4.
calculate
Linear Dependence of Vectors
A set of vectors is linearly independent if no vector in the set is:
1. a scalar multiple of another vector in the set or:
2. a linear combination of other vectors in the set
conversely, a set of vectors is linearly dependent if any vector in the set is:
1. a scalar multiple of another vector in the set
2. a linear combination of other vectors in the set.
Rank problem:
Does the Inverse Exist?
There are two ways to determine whether the inverse of a square matrix exists.
Determine its rank. The rank of a matrix is a unique number associated with a square matrix. If the rank of an n x n matrix is less than n, the matrix does not have an inverse.
Compute its determinant.
When the determinant for a square matrix is equal to zero, the inverse for that matrix does not exist.
A square matrix that has an inverse is said to be nonsingular or invertible; a square matrix that does not have an inverse is said to be singular.
Eksempel fra undervisning
