Intro to Linear Algebra Unit 1
Pivot position
entry that is a pivot of a row echelon form of that matrix
Vector equation
equation involving a linear combination of vectors with possibly unknown coefficients
Matrix equation
equation of the form Ax = b, where A is an m × n matrix, b is a vector in Rm, and x is a vector whose coefficients x1,x2,...,xn are unknown
Linear equation
equation on both sides have a sum of constant multiples of variables plus an optional constant multiple linear equations under a single brace is a system of linear equations solving the system means finding all solutions with formulas involving some number of parameters
Implicit equation
equation which relates the variables involved on one side system of equations contain implicit equations
Linear system of a certain number of equations and a certain number of unknown variables
equations determine amount of lines or planes unknown variables determines the dimension make up situations that can prove or disprove the questions
Gaussian elimination (row reduction)
every matrix is row equivalent to at least one matrix in reduced row echelon form swap the 1st row with a lower one so a leftmost non-zero entry is in the 1st row (if necessary) scale the 1st row so that its first non-zero entry is equal to 1 use row replacement so all entries below this 1 are 0 swap the 2nd row with a lower one so that the leftmost non-zero entry is in the 2nd row scale the 2nd row so that its first non-zero entry is equal to 1 use row replacement so all entries below this 1 are 0 swap the 3rd row with a lower one so that the leftmost non-zero entry is in the 3rd row use row replacement to clear all entries above the pivots, starting with the last pivot
Parametric solution set
every point on the line has the form (t, 1 − t) or (1 − t − w, t, w) for some real number t and collectively form the implicit equations for a line (x, y) = (t, 1 − t) or (x, y, z) = (1 − t − w, t, w)
Parameterized equation
expression that produces all points of the line in terms of one parameter
Pivot
first non-zero entry of a row of a matrix in row echelon form
Plane
flat sheet that is infinite in all directions
Equivalent
for any given list of vectors v₁, v₂, ..., vₙ, b, either all three statements are true, or all three statements are false
Row-column rule for matrix-vector multiplication
if A is an m × n matrix with rows r₁,r₂,...,rm, and x is a vector in Rn, then
Swap in an augmented matrix
interchange two rows
Matrix-Vector Product
let A be an m × n matrix, let u, v be vectors in Rⁿ, and let c be a scalar
Rⁿ
let n be a positive whole number set of all ordered lists of n real numbers n-tuple of real numbers is called a point of Rⁿ
Solution of a system of equations
list of numbers from variables that make all of the equations true simultaneously
Point
location in a plane drawn as a dot written horizontally
Spans and consistency
matrix equation Ax = b has a solution if and only if b is in the span of the columns of A this gives an equivalence between an algebraic statement (Ax = b is consistent), and a geometric statement (b is in the span of the columns of A)
Augmented matrix
matrix that consists of the coefficients and the constant terms in a system of equations m rows and n columns
Column vector
matrix with one column
Row vector
matrix with one row
Scalar multiplication
multiplication of a vector by a constant and can be positive or negative
Scaling in an augmented matrix
multiply all entries in a row by a non-zero number
Scaling in a system of equations
multiply both sides of an equation by a non-zero number
Vector multiplication
multiply each row in the second matrix by the column of the first matrix number of entries of x has to be the same as the number of columns of A
Reduced row echelon form
must first be in row echelon form each pivot is equal to 1 each pivot is the only non-zero entry in its column
Non-trivial solution
non-zero solution for a homogeneous system
Set builder notation
notation used to describe the elements of a set
Dimension
number of free variables in a solution set when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane
Row operation
operation performed on a row of an augmented matrix that creates an equivalent matrix
Vector subtraction
place the tail of v and w at the same point then v − w is the vector from the head of w to the head of v
Euclidean plane
plane consisting of an x-axis and y-axis with a ordered pair of coordinates x + y = 1
3D plane
plane consisting of an x-axis, y-axis, and z-axis with a ordered triple of coordinates x + y + z = 1
Vector
quantity that has magnitude and direction and not by location drawn as an arrow written vertically
Line
ray that is straight and infinite in both directions
Trivial solution
solution x = 0 of a homogeneous equation Ax = 0
Linear combination
sum of scalar multiples of vectors
Parallelogram Law for Vector Addition
sum of two vectors v and w is obtained as follows: place the tail of w at the head of v then v + w is the vector whose tail is the tail of v and whose head is the head of w doing this both ways creates a parallelogram
Swap in a system of equations
swap two equations
Consistent system of equations
system of equations that has one or more solutions corresponds to a consistent system of equations if each pivot corresponds to a value after the augmented line without a row of zeros on the bottom every column except the last column is a pivot column
Inconsistent system of equations
system of equations with no solution corresponds to an inconsistent system of equations if and only if the last column is a pivot column
Homogeneous system
system of linear equations of the form Ax = 0 when the homogeneous equation Ax = 0 does have nontrivial solutions, it turns out that the solution set can be conveniently expressed as a span
Inhomogeneous system
system of linear equations of the form Ax = b for b ≠ 0
Linearly independent
two vectors that are graphically on top of each other has vector solution x1v1+x2v2+···+xkvk=0 and has only the trivial solution x1=x2=···=xk=0
Three characterizations of consistency
vector b is in the span of v₁, v₂, ..., vₙ vector equation x₁v₁, x₂v₂, ..., xₙvₙ = b has a solution linear system with augmented matrix v₁, v₂, ..., vₙ is consistent
Row equivalent
when two matrices can be obtained from the other by doing some number of row operations
Infinite solutions
when two systems of equations intersect over each other last column is not a pivot column, and some other column is not a pivot column either
No solution
when two systems of equations never intersect last column is a pivot column
Parametric vector form
whenever a solution set is described explicitly with vectors but has free variables
Parametric form
write the system as an augmented matrix row reduce to reduced row echelon form write the corresponding solved system of linear equations move all free variables to the right hand side of the equations parametric form contains parameterized equations
Linearly dependent
Any collection of vectors containing the zero vector must be linearly dependent.
Row echelon form
all zero rows are at the bottom first non-zero entry of a row is to the right of the first non-zero entry of the row above below the first non-zero entry of a row, all entries are zero
Replacement in a system of equations
add a multiple of one equation to another, replacing the second equation with the result
Replacement in an augmented matrix
add a multiple of one row to another, replacing the second row with the result
Span of a single vector
all possible scalar multiples of that vector on a line
Free variable
any variable in a linear system that does not correspond to a pivot column augmented column is not free because it does not correspond to a variable free variables are independent variables whereas non-free variables are dependent if there are two free variables, only put one into parametric form and leave the others as is
Product of a row vector and column vector
a₁x₁ + a₂x₂ + ... + aₙxₙ
Weights
coefficients of a linear combination
Span
collection of all linear combinations and must contain the origin most span covers the entire plane but if there are free variables, they will only cover a line and zero vectors span will only cover the point on the origin in terms of spans, is (right of augmented matrix) in Span {(left of augmented matrix)}?
Solution set of a system of equations
collection of all solutions in a system of equations considered empty if there is no solution
Pivot column
column that contains a pivot position
Elimination method
combine the equations in various ways to try to eliminate as many variables as possible from each equation such as scaling, replacement, and swap
Vector addition
combining of vector magnitudes and directions