TRUE

Every elementary row operation is reversible.

FALSE

TRUE

A solution set "x" is a list of numbers "s" that makes each equation in the system a true statement when the values of "s" are substituted for "x" respectively 

TRUE

Two fundamental questions about linear systems involve existence and uniqueness

FALSE

Two matrices are row equivalent if they have the same number of rows

TRUE

Elementary row operations on an augmented matrix never change the solution set of the associated linear system

FALSE

Two equivalent linear systems can have different solution sets

TRUE

A consistent system of linear equations has one or more solutions

FALSE

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations

FALSE

The row reduction algorithm applies only to augmented matrices fora  linear system

TRUE

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix

FALSE 

TRUE

If a row in echelon form of an augmented matrix is [ 0 0 0 5 0], then the associated linear system is inconsistent

TRUE

FALSE

If every column of an augmented matrix contains a pivot, then the corresponding system is consistent

FALSE

The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process

TRUE

A general solution of a system is an explicit description of all solutions of the system 

FALSE

Whenever a system has free variables, the solution set contains many solutions.

FALSE

Another notation for vector -4

                                             3

is [-4 3]

FALSE

The points corresponding to -2

                                               5

and -5 lie on a line through the

           2  orgin

TRUE

An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1

TRUE

The solution of the linear system whose augmented matrix is

[a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b

TRUE

The set Span {u v} is always visualized as a plane through the orgin

FALSE

When u and v are nonzero vectors, Span {u v} contains only the line through u and the orgin, and the line through v and the orgin

TRUE 

Any list of five real numbers is a vector in R5

TRUE

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

FALSE

The vector v results when a vector u-v is added to the vector v

FALSE

The weights c1, ... cp in a linear combination c1v1 + ... + cpvp cannot all be zero

FALSE

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

TRUE

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

FALSE

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

TRUE

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

TRUE

If the columns of an m x n matrix A span Rm, then the equation Ax = b is consistent for each b in Rm 

FALSE

If A is an m x n matrix and if the equation Ax = b is inconsistent for some b in Rm, then A cannot have a pivot positon in every row 

TRUE 

Every Matrix equation Ax = b corresponds to a vector equations with the same solution set 

TRUE

TRUE

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

FALSE

If the coefficient matrix A has a pivot position in every row, then the equation Ax = b is inconsistent

TRUE

The solution set of a linear system whose augmented matrix is

[a1 a2 a3 b] is the same as the solution set of Ax = b if

A = [a1 a2 a3]

FALSE

Solution Set

the set of all possible solutions

(intersection between two lines: intersect, parallel, coincide at many points)

equivalent

two linear systems with the same solution set  

Row Equivalent

two matrices are this if there is a sequence of elementary row operations that transforms one matrix into the other (reversible)

Existent 

If the system is consistent it is said it is 

Unique

If there is only one solution set the solution is 

Echelon Form 

if matrix A is row equivalent to an echelon matrix U we call U _________ of A 

Pivot Position

a location in matrix A that corresponds to a leading 1 in the reduced echelon form of A

(have a nonzero number there)

Parametric Descriptions

descriptions of solution sets in which free variables act as parameters 

ordered pairs

vectors in R2 of real numbers

u + v

What corresponds to the forth vertex of the parallelogram whose other vertices are u, 0, and v

Linear combination

what is the vector defined by y in 

y = c1v1 + ... cpvp

called? 

(c1...cp are weights)

Linearly Independent 

has only the trivial solution

Linearly Dependent

if a set has weights that are not all zero the indexed set of vectors is...

Domain

FALSE

If A and B are 2x2 matrices with columns a1, a2, and b1, b2 respectively, then AB = [a1b1  a2b2]

FALSE

TRUE

AB + AC = A(B + C)

TRUE

A^T + B^T = (A + B)^T

FALSE

TRUE

The first row of AB is the first row of A multiplied on the right by B 

FALSE

If A and B are 3x3 matrices and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3] 

TRUE

If A is an n x n matrix, then (A^2)^T = (A^T)^2 

FALSE

(ABC)^T = C^T A^T B^T

TRUE

The transpose of a sum of matrices equals the sum of their transposes

TRUE

In order for a matrix B to be the inverse of A, the equations AB = I and BA = I  must both be true

FALSE

TRUE

TRUE

If A is an n x n matrix, then the equation Ax=b is consistent for each b is consistent for each b in Rn

TRUE 

FALSE

If A is invertible, then elementary row operations that reduce A to the indentity In also reduce A^-1 to In 

TRUE

If A is invertible, then the inverse of A^-1 is A itself

FALSE

A product of invertible n x n matrices is invertible, and the inverse of the product is the product of their inverses in the same order

TRUE

If A is an n x n matrix and Ax = ej is consistent for every j ∊{1, 2, ..., n}, then A is invertible. 

TRUE

If A can be row reduced to the identity matrix, then A must be invertible.