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Linear Algebra Test 1
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Gravity
Terms in this set (30)
span
the set of all linear combinations of a set of vectors
vector space
the space where linear combinations make sense; a set with two operations addition and scalar multiplication which satisfy 10 properties
subspace
the subset that is also a vector space
linear independent
a set is this if the only solution is a homogenous one
basis
a sequence of vectors on the vector space that span the space and are linearly independent
dimension
the number of elements in the basis
homogeneous linear system
a system of linear equation who's constants are zero in each equation
- will have either the trivial or infinite solutions
- swap rows
- scalar multiplication
- add rows
What are the elementary row operations?
- first nonzero entry in a row is one
- each leading 1 comes in a column to the right of the leading 1s in rows above it
- all rows of 0s come at the bottom of the matrix
- if a column contains at leading 1, them all other entries in the column are 0
How do you do reduced row echelon from?
-the first non-zero element in each row, called the leading entry, is 1.
-each leading entry is in a column to the right of the leading entry in the previous row.
-Rows with all zero elements, if any, are below rows having a non-zero element.
Steps for row echelon from
pivot positions
a location in a matrix A that corresponds to a leading 1 in the ref of A
pivot columns
column of A that contains a pivot
leading variables
left most nonzero entry in a nonzero
free variables
all other variables which are not the leading variables
- augment the matrix
- rref
What are the steps in Gauss-Jordan elimination?
- augment the matrix
- ref
- back substitue
What are the steps of Gaussian elimination?
identity matrix
a square matrix that, when multiplied by another matrix, equals that same matrix
non-singular
has an inverse and the determinant does not equal zero
singular
has no inverse and the determinant equals zero
diagonal matrices
zeros everywhere other then the main diagonal
triangular matrices
only contains values in the upper or lower part of the matrix and the rest is zeros
symmetric matrices
matrix reflects across the diagonal
-following any row or column of a determinant and multiplying each element of the row or column by its cofactor.
-the sum of these products equals the value of the determinant
How do you do calculate the determinant of a matrix by co-factor expansion?
take the products of the diagonal entries
How do you calculate the determinant of a diagonal matrix?
-adding = nothing happens
- scalar multiplication = matrix is a multiple of the
- row swapping = opp sign det
How does performing elementary row operations affect the determinate?
- find the det of the matrix A
-plug augmented column into row you solving for
- find the det of that matrix
- take the det x/ det of A
How do you use Cramer's rule to solve a linear system?
-closed under addition
- closed under scalar multiplication
- zero vector present
(- communicative
- associative
- distributive
- identity )
What is the gist of the 10 axioms of an abstract vector space?
row space
the span of the set of row
column space
the span of the set of columns
rank
the row rank( or column rank), the dim of the row(column) space, or number linearly in depended rows (columns)
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