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