True. Because replacement interchanging and scaling is all reversible.
Is the statement “Every elementary row operation is reversible” T or F
False because it has 5 rows
Is the Statement ” A 5×6 matrix has 6 rows” T or F
False because the description applies to a single solutin. the solution set consist of all possible solutions
Is the Statement” The solution set of a linear system involving variable x1-xn is a list of numbers (s1-sn) that makes each equation in the system a true statemnt when the values s1-sn are substituded for x1-xn respectfully” T or F
True because 2 fund questions address whether the solution exist and whether there is one solution..
Two fundamental questions about a linear system involve existence and uniquness T or F
false because if two matrices are row equivelent it means that there exists a sequence of row operations that transforms one matrix to the other
2 matrices are row equiv if they have the same number of rows
True because the elementary row operations replace a system with an equivelent system
elementary row operations on an augmented matrix never change the solution set of the associated linear system
false because 2 systems are called equivelent if they have the same solution set
2 equivelent linear systems can have different solution sets.
true consistent has atleast one solution
a consistent system of linear equations has one or more solutions
false becaue each matrix is equal to only one rre matrix
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 algorithm applies to any matrix whether augmented or not
the row reduction algorithm applies only to augmented matrices for a linear system
true it is the defenition of a basic variable
a basic variable in a linear system is a variable that corresponds to a pivot column in a coeffecient matrix
false the solution set of a linear system can only be expressed using a paremetric description if the system has atleast 1 solution
finding a parametric description of the solution set of a linear system is the same as solving the system
false because it doesnt make it inconstent all by itself
if 1 row in an echelon form of an augmented matrix is (0 0 0 5 0), then the associated linear system is inconsistent
false the echelon form is not unique but the reduced is
The echelon form of a matrix is unique
False the pivot position in a matrix are determined completly by the positions of the leading entries in the non zero rows of any echelon form obtained from the matrix
the pivot positions in a matrix depend on whether row interchanges are used in the row reduction process
True reducing a matrix to echelon form is called the forward phase and reducing to reduced echelon is called the backward phase
reducing a matrix to echelon form is called the forward phase of the row reduction process
false the existence of at least one solution is not related to free variables if the system is inconstent, the solution set is empty
whenever a system has free variables, the solution set contains many solutions
true the row reduction algorithm leads directly to an explict description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system
a general solution of a system is an explicit description of all solutions of the system
No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains infinitely many solutions. If the linear system is inconsistent, there is no solution
A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? Explain.
False. Span{u,v} includes linear combinations of both u and v.
When u and v are nonzero vectors, Span{u,v} contains only the line through u and the line through v and the origin.
True. set of real numbers R5 denotes the collection of all lists of five real numbers.
Any list of five real numbers is a vector in set of real numbers R5
False. Adding uminus−v to v results in u.
The vector v results when a vector uminus−v is added to the vector v.
The system is consistent because the rightmost column of the augmented matrix is not a pivot column
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent
False. The equation Axequals=b is referred to as a matrix equation because A is a matrix.
The equation Axequals=b is referred to as a vector equation. Choose the correct answer below.
True. The equation Ax=b has the same solution set as the equation x 1 Bold a 1 plus x 2 Bold a 2 plus times times times plus x Subscript n Baseline Bold a Subscript n Baseline equals Bold bx1a1+x2a2+•••+xnan=b.
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. Choose the correct answer below.
True. The first entry in Ax is the sum of the products of corresponding entries in x and the first entry in each column of A.
The first entry in the product Ax is a sum of products. Choose the correct answer below.
True. If the columns of A span set of real numbers R Superscript mℝm, then the equation Axequals=b has a solution for each b in set of real numbers R Superscript mℝm.
If the columns of an m times ×n matrix A span set of real numbers R Superscript mℝm, then the equation Axequals=b is consistent for each b in set of real numbers R Superscript mℝm. Choose the correct answer below.
True. If A is an m times ×n matrix and if the equation Axequals=b is inconsistent for some b in set of real numbers R Superscript mℝm, then the equation Axequals=b has no solution for some b in set of real numbers R Superscript mℝm.
If A is an m times ×n matrix and if the equation Axequals=b is inconsistent for some b in set of real numbers R Superscript mℝm, then A cannot have a pivot position in every row. Choose the correct answer below
True. The matrix equation Axequals=b is simply another notation for the vector equation x 1 Bold a 1 plus x 2 Bold a 2 plus times times times plus x Subscript n Baseline Bold a Subscript n Baseline equals Bold bx1a1+x2a2+•••+xnan=b, where Bold a 1a1, …, Bold a Subscript nan are the columns of A.
Every matrix equation Axequals=b corresponds to a vector equation with the same solution set. Choose the correct answer below.
True. The equation Axequals=b has a nonempty solution set if and only if b is a linear combination of the columns of A.
If the equation Axequals=b is consistent, then b is in the set spanned by the columns of A. Choose the correct answer below.
True. The matrix A is the matrix of coefficients of the system of vectors.
Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. Choose the correct answer below.
False. If A has a pivot position in every row, the echelon form of the augmented matrix could not have a row such as [0 0 0 1], and Axequals=b must be consistent.
If the coefficient matrix A has a pivot position in every row, then the equation Axequals=b is inconsistent. Choose the correct answer below.
No. The matrix A whose columns are the three vectors has four rows. To have a pivot in each row, A would have to have at least four columns (one for each pivot.)
Could a set of three vectors in set of real numbers R Superscript 4ℝ4 span all of set of real numbers R Superscript 4ℝ4? Explain. Choose the correct answer below.
No. The matrix A whose columns are the n vectors has m rows. To have a pivot in each row, A would have to have at least m columns (one for each pivot.)
Could a set of n vectors in set of real numbers R Superscript mℝm span all of set of real numbers R Superscript mℝm when n is less than m? Explain. Choose the correct answer below.
