Math 1553 final exam

When I mention a ‘2 × 3 matrix’, the first number indicates the (row/column) and the second umber indicates the (row/column)

row, column

a consistent system means ____, and an inconsistent system means ______.

at least one solution exists for the system, a solution does not exist for the system

R is a Euclidean _____, R² is a Euclidean _____, R³ is a Euclidean ____.

line, plane, space

what is the implicit form of a linear equation?

the implicit form is the regular looking way of the equation

what are examples of non linear equations?

where variables are multiplied to each other instead of being added or subtracted to each other

what are the three types of solutions a linear system of equations can have?

no solution, 1 solution or infinitely many solutions

what does it mean to have m linear equations in n unknowns in terms of rows and columns?

there are m linear equations → rows, and n variables → columns

do augmented matrices initially have 0’s to the right of the separatrix or numbers?

numbers

break it down: what does it mean to write an element of Rⁿ ? tell me what an element is, and what Rⁿ is?

an element is a member of a set (a set is a collection of things where order does not matter). Rⁿ is the set of all vectors, with n representing the number of entries within each vector.

what would you write if you were told to write an element of Rⁿ?

write a list with elements within it, and have each entry represent the given situation

what is a pivot in REF and RREF?

REF: a number you will sense will be 1 in RREF with 0’s above and below it

RREF: what I described above

what is a matrix in row echelon form?

1) all 0 rows are at the bottom

2) each leading entry in a row is to the right of the leading entry in a row above

3) directly below a leading entry, all entries are 0

what is a matrix in row echelon form?

it’s in row echelon form and the pivots are equal to 1, and each pivot is the only non zero entry in its column

true or false: a linear system is inconsistent if the augmented matrix’s right most column is a pivot column

true

if there is 1 free variable, then the solution set of the system of linear equations is a ____.

line

what does R³ mean?

all possible points in a 3-d space; (x, y, z)

true or false: 2 planes in R3 can intersect at exactly one point? If false, state the correct answer.

false; 2 planes in R3 can only intersect at a line or not at all

if a solution to a system of linear equations has 2 free variables, then the solution set is a ____.

plane

in terms of pivots and columns, what are the conditions for a matrix to have no solution, one solution or infinitely many solutions?

for no solution, the right most column of any matrix has a pivot

for one solution, every column except the right most column has a pivot

for inf. many solutions, the right most column doesn’t have a pivot and some other column doesn’t either

define what a point is in terms of Rⁿ

a point is an element (or a vector) of Rⁿ (a single vector in a set of vectors of Rⁿ)

how would you write an element of Rⁿ on a graph?

write it as an (x.y) pair on the graph

define what a vector is in terms of Rⁿ

an element of Rⁿ drawn as an arrow

do vectors always start at the origin?

no

how would you draw (2) as a vector on a graph?

(1)

start from wherever you wanna start and draw an arrow to the point (2,1) on the graph

how do you add 2 vectors together?

just add them straight across from each other to make a new vector

how do you scale a vector by a scalar?

multiply each entry by the scalar to get a new vector

how do you draw the addition and subtraction of two vectors geometrically?

for addition, you would take the vector line from one vector and add it to the other vector, adding it in a positive direction, and draw a vector line to the new point where the addition of the two vectors end. For subtraction, you would do a similar thing, but add the other vector going in a negative direction onto the other vector.

what happens if you scale a vector geometrically by a scalar? what happens if you negate a vector geometrically?

the same vector either gets shorter or longer depending on if you multiply the vector by a scalar greater than 1 or less than 1 for scaling a vector by a scalar. You make the vector go in the opposite direction if you negate it.

w vector + ( v vector - w vector) =

v vector

what makes a vector a linear combination? what does this mean intuitively?

a vector w in Rⁿ is a linear combination of the vectors v₁, …, v_p in Rⁿ if w can be written as w = c₁v₁ + c₂v₂ + … +c_pv_p for some scalars c₁, …, c_p.

intuitively, we can get to w by taking c₁ steps (telling you how far to go in the direction) in the v₁ direction and c₂ steps in the v₂ direction and … taking c_p steps in the v_p direction.

what is the Span {v₁,…v_p} mean?

it’s the set of all vectors of the form: {x₁v₁ + … x_p+v_p}, with x being real numbers

true or false: the Span of {v₁, …, v_p} is the set of all linear combinations

true

write the equation to the vectors of Span {v₁, v₂}

x₁v₁ + x₂v₂

what has to be true for the entries of v1 and v2 for Span {v1, v2) = R²

both vectors have to have 2 entries each

true or false: Span {v1, v2) = R² is always a plane

false, it can be a plane or a line

the Span of the 0 vector is ….

a point

true or false: a system is consistent only if the vector equation is consistent (meaning there exists a solution)

true

true or false: in an augmented matrix, without solving for the variables, b is a linear combination for the vectors and b is in Span of those vectors

true

how do you write a vector equation from a given system of equations ?

separate them into vectors by variable, and have b as a vector by itself as well. then do parametric form, and leave b alone. then for the vector equation, have it in the form x₁v₁ + x₂v₂ = b, and have each vector be represented as v₁ or whatever in the vector equation. then once you solve for x and y using row reduction, plug in the x and y values into the vector equation. you can also plug in the actual vectors into the vector equation as well, including the b vector.

how to write a set of different vectors whose span is a line in R³?

provide vectors, each with 3 entries, with all of them being linearly dependent of each other (being scalar multiples of each other).

how to write a set of different vectors whose span is a plane in R³?

give two independent vectors and, and write the third vector so that it is a linear combination of the first 2 vectors

how to provide a set of different vectors who span is only a single point in R³?

0 vector for all

what does Ax = b mean?

multiplying the columns of A by the vector of unknowns give me the destination, b

true or false: saying the following all mean the same thing:

b is in Span {v1,…,vp}, with b being a linear combination

the vector equation is consistent

the augmented matrix is consistent

the matrix Ax = b is consistent

true

n in Rⁿ is equal to the number of

rows

the number of columns in a matrix is equal to the number of ____ ____ in the vector equation

vector coefficients

true or false: the columns of A are the same as x vector in Rⁿ

true

each column in A lives in R^_?

m

true or false;

Ax = (v₁, v₂, …., v_p) (x₁) = the vector equation

(x₂)

(. ..)

(x_p)

true

Ax lives in R^_?

m

what is the condition for multiplying a matrix by a vector?

the number of columns of the matrix must match up with the number of rows of the vector

how to multiply a matrix by a vector?

(first entry of the vector x the first column of A) + (second entry of the vector x the second column of the matrix) and so on… * end up with a vector

what is a row vector?

a vector that is only one row

true or false: (a₁ …. a_n) (x₁) = a₁x₁ + … + a_nx_n

(…)

(x_n)

true

for a row vector, m is the number of ____ and n is the number of _____

rows, columns

if A m x n w/ rows r₁, …, r_m, and if x in Rⁿ: Ax =

(r1)

(r2) ( x)

(…)

(r3)

if A m x n w/ rows r₁, …, r_m, and if x in Rⁿ, where does Ax live in?

R^m

for an m x n matrix, the columns of A live in ____, the vector x lives in ____, the output lives in _____ .

R^m, Rⁿ, R^m

true or false: if A m x n, b in R_m, then Ax = b has a solution x.

true

true or false: if b is in Span {v₁, …,v_n}, then b is in the span of the columns of A

true

A m x n, b in R^m

What conditions on A guarantees that Ax = b consistent, no matter what b is?

A has a pivot in evert row

for Ax = b, it is a linear system of ____ equations in ___ unknowns

m, n

A ( u + v ) =

Au + Av

A (cu) =

cAu

the solution set to Ax = 0 is always the span of

the vectors you get from the free variables

true or false: every vector in Rⁿ being a linear combination of the columns of A guarantees that Ax = b is consistent for every b in Rⁿ

^{what does this statement mean?}

true.

that no matter what b you pick, you can still write the vector equation

in order to guarantee that Ax = b is consistent for every b in Rⁿ, the set of solutions to Ax = 0 has to be a what?

whatever shape will give the matrix a pivot in every row

define what a homogenous and a non homogenous system is?

homogenous is Ax = 0

non homogenous is Ax = b, with b not being equal to 0

every homogenous linear system is consistent when Ax = 0, when x =

0

what is the trivial solution?

x = 0

Ax = 0 has inf. many solutions if it has at least 1 ____

free variable

how to find the solution set to Ax = 0?

row reduce to RREF and parametric vector form and its the span of those columns

how to find the solution set to Ax = b, with b being a vector other than the 0 vector?

augment the matrix to have the right most column be b, RREF and put in parametric vector form and its the span of those columns

what is parametric vector form?

getting the columns with the free variables

true or false:

after getting the solution in parametric vector form if you’re left with a vector multiplied to a free variable and a vector not multiplied by a free variable, then this means the solution set is a translation of the span (the vector with the free variable) by the vector w/o the free variable

this also means that the solution set is a line through the vector w/o the free variable, parallel to the Span of the vector w/ the free variable

also, the vector w/o the free variable itself is a solution to Ax = b, where b is not the 0 vector

true for all

true or false:

the solution set to a consistent Ax = b is obtained by taking any particular solution to Ax = b and adding all solutions to Ax = 0

true

true or false: if you have Ax = b, and you solve the same equation for two different b’s, then the solution sets from both b’s will be parallel to each other

define what a solution set is in terms of Ax = b

define what the columns span of A is

1) for fixed b, all x so that Ax =b

2) all b so that Ax = b is consistent

the span of the columns of A are …

the span of the columns of A as vectors

if Span {u, v} is a plane, they are linearly _____. If Span {u,v} is a line or point, they are linearly ______.

independent, dependent

what is the definition for a set of vectors being linearly independent or linearly dependent?

a set of vectors {v₁,…v_p} in Rⁿ is linearly independent if the vector equation x₁v₁, + …., x_pv_p = 0 has only the trivial solution, where all coefficients of the vector equation are 0.

Otherwise, the set of vectors {v₁,…v_p} are linearly dependent, where at least one of the coefficients in the vector equation is not 0.

true or false:

saying the set of vectors {v₁,…,v_p} in Rⁿ is lin. inde. if the vector equation x₁v₁ +…_x_pv_p = 0 has only the trivial solution is the same as saying (v₁…v_p) (x₁…x_p) = 0 has only the trivial solution, where x₁ = 0, x₂ = 0,…, x_p=0, which is the same as saying Ax = 0 has only the trivial solution x = 0, where Ax = 0 has no free variables and has a pivot in every column.

true

How do I check to see if vectors are lin. ind?

row reduce and if every column has a pivot, then they are lin. ind.

is the set of vectors {v₁,...,v_p} lin. ind. or dep.?

dep

what does it mean for a set of vectors {v₁,…v_p} to be lin. dep.?

at least one of the vectors in the span of the others, so you can remove at least one vector without shrinking the span

what is the increasing Span criterion

{v₁,…v_p} is lin. ind. if you add one of these vectors to the set and it increases the Span of these vectors

{v₁} is lin. ind. if Span {v₁} is a ____.

{v₁,v₂} is lin. ind. if Span {v₁,v₂} is a ____.

{v₁,v₂,v₃} is lin. ind. if Span {v₁,v₂,v₃} is a _______.

line, plane, 3-d space

the pivot columns of a matrix are lin. _____

ind.

the columns of a matrix are lin. ind. if A has a pivot in every _____

column

matrices that have more columns than rows are lin. ____

dep.

if the columns of a matrix are lin ind., then Ax = b has either _____ solution or _____ solution

one, no

true or false:

the xy-plane in R³ is like R² but it’s not R²

the line thru the 0 vector in R10 is like R but it’s not R

Rⁿ is like itself

true

list the three criteria for a subspace V to be a subspace of Rⁿ

1) the 0 vector is in V

2) if u,v in V, then u+v is in V (closed under addition)

3) if u in V, then cu is in V (closed under scalar multiplication)

true or false:

every span of vectors is a subspace and every subspace is in the span of some vectors

true

what is the definition of a spanning set for V?

what does it mean that V is spanned by v₁,..,v_p?

if V = Span {v₁,…,v_p}, V is the subspace spanned by v₁,…,v_p, and v₁,…,v_p is a spanning set for V.

what is col A and nul A? where do they live? how do you solve for each?

col A, or the column space of A, is the span of the columns of A, and lives in R^m. You find a spanning set of col A by RREF the matrix, and the same columns from the original matrix are the spanning set for col A from the pivot columns from the RREF matrix. It’s the span of those columns.

nul A, or the null space of A, is the solution set to Ax = 0, and it lives in Rⁿ. You find the nul A by RREF and finding parametric vector form, and the span of those columns is the nul A.

define what a basis for V is

let V be a subspace of Rⁿ:

a basis of V is a set {v₁,…,v_p}, where

1) {v₁,…,v_p} is lin. ind.

2) Span {v₁,…,v_p} = V

define what the dimension of V is

the number of vectors in basis for V

true or false: a subspace for V usually has inf. many choices for a basis for V

true

every basis for V has the same number of ______.

vectors