Linear Algebra: Matrices Flashcards
Linear System
A finite set of linear equations in the variables x1, x2, … xn
Inconsistent linear system
A system with no solutions.
Consistent linear system
A system with at least one solution.
3 Elementary row operations
- Multiply a row throughout by a nonzero constant - Interchange two rows - Add a multiple of one row to another
Row Echelon form
A matrix with properties: - The first nonzero number in every row is a 1 (leading 1). (Except if a row consists entirely of zeros). - Any rows consisting entirely of zeros are grouped together at the bottom of the matrix. - The leading 1 in the lower row occurs further to the right than the leading row in the higher row (in non-zero rows).
Reduced Row Echelon form
A matrix with all the properties of Row Echelon form and: - Each column that has a leading 1 has zeros everywhere else in that column.
General Solution
A set of parametric equations from which all solutions of a linear system can be obtained by assigning numerical values to the parameters.
Homogeneous
A system of linear equations where the constant terms are all zero.
Trivial solution to a homogeneous system
x1 = x2 = …. = xn = 0
Nontrivial solution to a homogeneous system
Any solution other than the trivial solution
2 Possibilities for any homogenous linear system:
- Only the trivial solution - Infinitely many solutions, including the trivial solution.
Number of solutions to a homogenous system with more unknowns than equations.
Infinite
Number of free variables in a homogenous linear system that has n unknowns, and a reduced row echelon form with r nonzero rows.
n-r
Pivot positions
The positions of the leading 1’s.
Matrix
A rectangular array of numbers - called entries.
Column matrix
A matrix with only one column
Row matrix
A matrix with only one row
Scalars
Numerical quantities
Equal matrices
Matrices with the same size, whose corresponding entries are equal.
Transpose of a matrix
A matrix of the same size as the original, obtained by interchanging the rows and columns of A.
Trace
The trace of a square matrix, denoted tr(A), is the sum of the entries on the main diagonal of the matrix.
Zero matrix
Any matrix with only zero entries.
Identity matrix
Square matrix with 1’s on the main diagonal and zero’s elsewhere.
Explain an inverse
If A is a square matrix, and B a matrix of the same size, such that AB = BA = I, then A is said to be invertible and B is called an inverse of A.
Singular matrix
A matrix that has no inverse.
Nonnegative integer powers of a matrix (A)
A0 = I
An = AA … A (n times, n>0)
Negative integer powers of a matrix (A)
A-n = ( A-1)n = A-1A-1… A-1 (n times)
Exponential Laws of matrices
If A is a square matrix and r and s are integers then:
- ArAs = Ar+s
- ( Ar)s = Ars
Define constant a in matrix terms
a.I
5 Properties of the transpose of a matrix
- ( (A)T)T = A
- ( A+B )T = AT + BT
- ( A-B )T = AT - BT
- ( kA )T = kAt, k being any scalar
- ( AB )T = BTAT
Elementary matrix
A matrix that can be obtained from the nxn identity matrix In by performing a single elementary row operation.
Theorem 1.5.3 on matrices (4 all true/all false)
- A is invertible
- Ax = 0 has only the trivial solution
- The reduced row echelon form of A is In
- A is expressible as a product of elementary matrices
Row equivalent matrices
Matrices that can be obtained from one another by a finite sequence of elementary row operations.
Method for inverting matrices
We must find a sequence of elementary row operations that reduces A to the identity and then perofrm this same sequence of operations to In to obtain A-1.
