Linear Algrebra Flashcards
(29 cards)
Vector space
A (nonempty) set V of vectors such that with any two vectors a and b in V all their linear combinations are elements of V, and all these vectors satisfy the laws for matrix addition and scalar multiplication
Transpose
A matrix in which the rows and columns have interchanged
Symmetric matrix
When a matrix is equal to its transpose
Antisymmetric matrix
When a matrix is equal to the negative of its transpose
Diagonal matrix
Square matrix where all elements not in the leading diagonal are 0
Identity matrix
A diagonal matrix with all elements 1
Inverse matrix
The matrix then when multiplied by the original matrix equals the identity matrix
Minor
The matrix obtained by removing all the elements in the chosen element’s rows and columbs
Cofactor
Multiply the minor by (-1)^(i+j)
The determinant
The sum of the products of the elements of any row or column and their corresponding cofactors
Properties of determinants
-determinant of the transpose
-interchanging 2 rows or columns
-removing factors
-add a constant multiple of one row (column) to another
-identical rows or columns
-determinant of a product
Determinant of the transpose
The transpose of A has the same determinant as A
Interchanging two rows or columns
If two rows (columns) are interchanged, its determinant changes sign but not magnitude
Removing factors
If all elements in a single row have a common factor, then this factor can be removed. The value of the determinant is given by is given by the product of the remaining determinant and the factor removed
Adding a constant multiple of one row/column to another
The determinant of a matrix is unchanged in value by adding to the elements of one row (column) to any mixed multiple of the elements of another row (column)
Identical rows or columns
If any two rows or columns are identical or multiples of each other, then the determinant is 0
Determinant of a product
If A and B are square matrices of the same order then the products of the determinant is equal to the determinant of their product
Singular matrix
A matrix that has a determinant of 0
Linear dependence
The determinant of the matrix is zero if the rows (columns) are linearly dependent
The adjoint matrix
The transposed matrix of cofactors
Inverse matrix
Adj A/ det A
Orthogonal matrices
A square matrix thats inverse equals its transpose.
Eigenvalue
A value that satisfied det(A -yI) = 0
Eigenvectors
Solutions to Ax = yx. If A is a nxn matrix it will have at most n Eigen vectors
