Linear 1

Suppose that a system of six equations with fourteen unknowns corresponds to a matrix in row echelon form. What is the largest possible number of pivots for this matrix?

6

linearity property of matrixes

A (u + v) = Au + Av A(cu) = cAu if c scaler

A mxn, b in R^m. What condition on A guarantees that Ax=b will be consistent

A has a pivot in every row

If an augmented matrix in RREF has a pivot in the rightmost column, then the corresponding systems of linear equations must be inconsistent

T

The equation 3x+ln⁡(2)y=π is a linear equation in x and y.

T

Ax = b is consistent if and only if

b is in the column span of A

point

element of R^n drawn as a dot

vector

element of R^n drawn as an arrow usually start at the origin

getting RREF

from left to right, identify pivots one by one and eliminate all entries below pivots from right to left eliminate all entries above the pivots (somewhere in the process scale so that each pivot is 1 and make sure zero rows are at bottom)

3 possibilities for solution sets of linear equations based on augmented matrix

no solution: right most column of augmented matrix has a pivot 1 0 0 0 0 0 0 1 exactly one solution: every column except the rightmost column has a pivot 1 0 0 1 0 1 0 2 0 0 1 3 0 0 0 0 infinitely many solutions: rightmost column doesn't have a pivot, and some other column doesn't either 1 0 0 7 0 1 0 -3

Take S, xy-plane in R^3 Is S equal to R^2

NO because every point in R^2 has exactly 2 coordinates (x,y) but every point in S has 3 coordinates but the third coordinate is 0. (x,y,0). S is like R^2 but not equal to R^2

Is it possible for 2 planes in R^3 to intersect at exactly one point

No, if it is consistent you will have a free variable and infinitely many solutions. If inconsistent then you will have no solution

1 free variable 2 free variables no free variables

line plane point

RREF or not? 0 0 1 0

no

RREF or not? 1 -1 0 4 0 0 0 1

no

Determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions. 1 0 0 2 0 1 0 4 0 0 0 1

no solution

possibilities for solutions to 2 linear equations in 2 unknowns x, y:

no solution (inconsistent): x+y=2 x+y=1 exactly one (consistent): x-y=2 x+y=4 infinitely many solutions: x-y=0 10x-10y=0

Suppose we are given a linear system of 3 equations in the 2 variables x and y. Which of the following are possible for the number of solutions for the system? no solution one solution two solutions three solutions infinitely many solutions

no solution: yes one solution: yes two solutions: no three solutions: no infinitely many solutions: yes

RREF or not? Consistent or inconsistent? 0 1 2 2 3 0 0 0 1 0

not RREF consistant

vector equations xv1 + yv2 = b

our system of equations is consistent if and only if the vector equation involved is consistant b= linear combination of v1 and v2 b= span {v1, v2} the matrix equation is consistent

4 ways to write linear systems

system: 2x-y=5 x+4y=7 augmented matrix: 2 -1 5 1 4 7 vector equation: x 2 + y -1 = 5 1 4 7 matrix equation: 2 1 x = 5 1 4 y 7

a linear system is inconsistent if and only if

the augmented right most column is a pivot column

Ax=0 has infinitely many solutions if

there is at least 1 free variable and A has at least one column with no pivot

linear combination

A vector w in R^n is a linear combination of the vectors v1...vp in R^n if w can be written: w = cv1 + c2v2 ... + cpvp for some scalars c1, ..., cp

every homogeneous system is consistent

Ax=0 when x=0 call x=0 trivial solution

If a linear system has four equations and seven variables, then it must have infinitely many solutions.

F

If the bottom row of an augmented matrix in reduced row echelon form is (0,1,3,1), then the system has no solution.

F

If the number of rows of an augmented matrix in reduced row echelon form is greater than the number of columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.

F

If x is a nontrivial solution of Ax=0, then every entry of x is nonzero.

F

Span{a1,a2} contains only the line through a1 and the origin, and the line through the a2 and the origin.

F

The equation Ax=b is consistent if the augmented matrix [A∣b] has a pivot position in every row.

F

The equation Ax=b is referred to as a vector equation.

F

The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.

F

The plane 2x+y−z=3 contains the point (1,1,1).

F

The set of all solutions (x,y,z) to the equation x−y−z=0 is a line in R3.

F

The solution set of a linear system whose augmented matrix is [a1a2a3∣b] is the same as the solution set of Ax=b, if A=[a1a2a3].

F

There are exactly three vectors in Span{a1,a2,a3}.

F

Determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions. 1 0 0 1 0 1 0 3 0 0 0 0

Infinitely many solutions

RREF or not? Consistent or inconsistent? 1 0 0 7 0 0 0 0 0 1

RREF inconsistant

A homogeneous linear system is always consistent.

T

A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.

T

Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.

T

Asking whether the linear system corresponding to an augmented matrix [a1a2a3b] has a solution amounts to asking whether b is in Span{a1,a2,a3}.

T

E. The first entry in the product Ax is a sum of products.

T

If the bottom row of an augmented matrix in RREF ( 0 1 0 0) then the corresponding systems of linear equations has infinitely many solutions

T

If the solution to a system of linear equations is given by (4−2z,−3+z,z), then (4,−3,0) is a solution to the system.

T

The equation Ax=b is homogenous if the zero vector is a solution.

T

The solution set of a consistent inhomogeneous system Ax=b is obtained by translating the solution set of Ax=0.

T

The solution set of the linear system whose augmented matrix [a1a2a3b] is the same as the solution set of the equation x1a1+x2a2+a3x3=b.

T

There are exactly three vectors in the set {a1,a2,a3}.

T

Let A= 4 4 2 11 2 4 We want to determine if the system Ax=b has a solution for every b∈R3. There is a solution for every b in R3 but we need to row reduce A to show this. There is a not solution for every b in R3 but we need to row reduce A to show this. There is not a solution for every b in R3 since 2<3. There is a solution for every b in R3 since 2<3 We cannot tell if there is a solution for every b in R3.

There is not a solution for every b in R3 since 2<3.

Determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions. 1 0 0 0 1 0 0 0 0

Unique solution: x=0,y=0

Determine whether it has a unique solution (in this case, find the solution), infinitely many solutions, or no solutions. 1 0 0 0 0 1 0 1 0 0 1 -1

Unique solution: x=0,y=1,z=−1

What describes the system x-5y+7z=1 2y-2z=2 3y-3z=3

a line in R^3

a matrix is in reduced row echelon form if it is in row echelon form and

all poivot = 1 each pivot is the only non zero entry in its colummn 1 0 0 0 1 0 0 0 1 1 4 0 -1 0 0 1 15

A matrix is in row echelon form if it satisfies

all zero rows are at the bottom each leading entry in a row is to the right of leading entry in row above directly below a leading entry, all entries are 0 3 1 9 5 42 0 1 3 0 1 0 0 0 4 13 0 0 0 0 0

solutions to Ax=0 is

always the span of some number of vectors

RREF or not? 0 0 0 0

yes

RREF or not? 0 0 1 0 0 0 0 0

yes

RREF or not? 1 0 2 0 1 -1

yes

examples of non linear equations

√x, x^2, siny, ln(z), xy