Linear Algebra Test 1

Basic variable

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

One-to-one

A mapping T is said to be one-to-one if each b in R^m is the image of at most one x in R^n T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution

C.

Answer on reverse side

(a) yes (b) yes

Answers on reverse side

Matrix B has 3 rows.

How many rows does B have if BC is a 3 x 5 matrix?

Random Definitions

Random Definitions

Section 1.6: Balancing Chemical Equations and Flow Problem

Review how to balance chemical equations and do flow problems

In some​ cases, a matrix may be row reduced to more than one matrix in reduced echelon​ form, using different sequences of row operations. Is this statement true or​ false? The statement is false. For each​ matrix, there is only one sequence of row operations that row reduces it. The statement is true. The echelon form of a matrix is always​ unique, but the reduced echelon form of a matrix might not be unique. The statement is true. It is possible for there to be several different sequences of row operations that row reduces a matrix. The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.

The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.

What must be true of a linear system for it to have a unique​ solution? Select all that apply. The system has exactly one free variable. The system is consistent. The system has at least one free variable. The system has one more equation than free variable. The system has no free variables. The system is inconsistent.

The system is consistent. The system has no free variables.

False. Adding u-v to v results in u.

The vector v results when a vector u-v is added to the vector v. True. Adding u-v to v results in v. False. Adding u-v to v results in u-2v. False. Adding u-v to v results in 2v. False. Adding u-v to v results in u.

False. Setting all the weights equal to zero results in the vector 0.

The weights c 1, ... ,c_p in a linear combination c_1*v_1 + ... + c_p*v_p cannot all be zero. False. Setting all the weights equal to zero results in the vector 0. True. Setting all the weights equal to zero results in the vector 0. True. Setting all the weights equal to zero does not result in the vector 0. False. Setting all the weights equal to zero does not result in the vector 0.

If A is an m x n matrix and if the equation Ax=b is inconsistent for some b in set of real numbers R^m, then A cannot have a pivot position in every row. Choose the correct answer below. False. If A is an m x n matrix and if the equation Ax=b is inconsistent for some b in set of real numbers R^m, then the equation Ax=b has a solution for each b in set of real numbers R^m. True. If A is an m x n matrix and if the equation Ax=b is inconsistent for some b in set of real numbers R^m, then the equation Ax=b has no solution for some b in set of real numbers R^m. False. Since the equation Ax=b has a solution for each b in set of real numbers R^m, the equation Ax=b is consistent for each b in set of real numbers R^m. True. If A is an m x n matrix and if the equation Ax=b is inconsistent for some b in set of real numbers R^m, then the columns of A span set of real numbers R^m.

True. If A is an m x n matrix and if the equation Ax=b is inconsistent for some b in set of real numbers R^m, then the equation Ax=b has no solution for some b in set of real numbers R^m.

Sections 1.1 and 1.2

True/False questions from Section 1.1 and 1.2 Homework

Section 1.4

True/False type questions from homework over section 1.4 switch back to term first

Section 1.8

True/False type questions from homework over section 1.8

Is the statement​ "A 5x6 matrix has six​ rows" true or​ false? Explain. ​True, because a 5x6 matrix has five columns and six rows. ​True, because a 5x6 matrix has six columns and six rows. ​False, because a 5x6 matrix has five rows and five columns. ​False, because a 5x6 matrix has five rows and six columns.

​False, because a 5x6 matrix has five rows and six columns.

The columns of a matrix A are linearly independent if and only if Ax=0 has no free​ variables, meaning every variable is a basic​ variable, that​ is, if and only if every column of A is a pivot column.

(a) Fill in the blank. If A is an m x n ​matrix, then the columns of A are linearly independent if and only if A has n pivot columns. Why is the statement in​ (a) true? The columns of a matrix A are linearly independent if and only if Ax=0 has no free​ variables, meaning every variable is a basic​ variable, that​ is, if and only if every column of A is a pivot column. The columns of a matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution. This happens if and only if Ax=0 has no free​ variables, meaning every variable is a basic​ variable, that​ is, if and only if there are 0 pivot columns. The columns of a matrix A are linearly independent if and only if the equation Ax=0 has more unknowns than equations so there must be free variables. This happens if and only if there are m pivot columns.

Onto transformation

A transformation is onto if the standard matrix has a pivot in every row. A mapping T: R^n -> R^m is said to be onto R^m if each b in R^m is the image of at least one x in R^n Equivalently, T is onto R^m when the range of T is all of the codomain of R^m. That is, T maps R^n onto R^m if, for each b in the codomain R^m, there exists at least one solution of T(x)=b

True. set of real numbers R^5 denotes the collection of all lists of five real numbers.

Any list of five real numbers is a vector in R^5. False. A list of numbers is not enough to constitute a vector. False. A list of five real numbers is a vector in set of real numbers R^6. True. set of real numbers R^5 denotes the collection of all lists of five real numbers. False. A list of five real numbers is a vector in set of real numbers R^n.

True. An augmented matrix has a solution when the last column can be written as a linear combination of the other columns. A linear system augmented has a solution when the last column of its augmented matrix can be written as a linear combination of the other columns.

Asking whether the linear system corresponding to an augmented matrix [ a_1 a_2 a_3 b] has a solution amounts to asking whether b is in Span {a_1,a_2,a_3}. False. An augmented matrix having a solution does not mean b is in Span {a_1, a_2, a_3}. True. An augmented matrix has a solution when the last column can be written as a linear combination of the other columns. A linear system augmented has a solution when the last column of its augmented matrix can be written as a linear combination of the other columns. False. If b corresponds to the origin then it cannot be in Span {a_1, a_2, a_3}.

The equation Ax=b is consistent if the augmented matrix [ A b ] has a pivot position in every row. Choose the correct answer below. False. The augmented matrix [ A b ] cannot have a pivot position in every row because it has more columns than rows. False. If the augmented matrix [ A b ] has a pivot position in every​ row, the equation equation Ax=b may or may not be consistent. One pivot position may be in the column representing b. True. If the augmented matrix [ A b ] has a pivot position in every​ row, then the equation Ax=b has a solution for each b in set of real numbers R^m. True. The pivot positions in the augmented matrix [ A b ] always occur in the columns that represent A.

False. If the augmented matrix [ A b ] has a pivot position in every​ row, the equation equation Ax=b may or may not be consistent. One pivot position may be in the column representing b.

The equation Ax=b is referred to as a vector equation. Choose the correct answer below. False. The equation Ax=b is referred to as a linear equation because b is a linear combination of vectors. True. The equation Ax=b is referred to as a vector equation because A is constructed from column vectors. False. The equation Ax=b is referred to as a matrix equation because A is a matrix. True. The equation Ax=b is referred to as a vector equation because it consists of scalars multiplied by vectors.

False. The equation Ax=b is referred to as a matrix equation because A is a matrix.

False. The alternative notation for a​ (column) vector is ​(-4​,3​), using parentheses and commas.

False. The matrices are not equal because they have different​ entries, even though they have the same shape. True. The matrices are equal because they have the same​ entries, even though they have different shapes. False. The alternative notation for a​ (column) vector is ​(-4​,3​), using parentheses and commas. True. The matrices both represent the geometric point ​(-4​,3​).

If a set contains fewer vectors than there are entries in the​ vectors, then the set is linearly independent. Choose the correct answer below. True. If a set contains fewer vectors than there are entries in the​ vectors, then there are less variables than​ equations, so there cannot be any free variables in the equation Ax=0. False. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the​ vectors, then at least one of the vectors must be written as a linear combination of the others. False. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector. True. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector.

False. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.

Linearly independent

If A is an mxn matrix, then the columns of A are linearly independent if and only if A has n pivot columns

The size of B is 6 x 5 A is an m x n matrix B is an n x p matrix so AB is a m x p matrix

If a matrix A is a 8 x 6 and the product AB is 8 x 5, what is the size of B?

Linear dependence

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent If a set S = {v_1, ... , v_p} in R^n contains the zero vector, then the set is linearly dependent A set of two vectors is linearly dependent if at least one of the vectors is a multiple of the other

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. No, it cannot have a unique solution. Because there are more variables than​ equations, there must be at least one free variable. If there is a free​ variable, the solution set contains a unique solution. ​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. ​Yes, it can 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 either a unique solution or infinitely many solutions. If the linear system is​ inconsistent, there is no solution. ​Yes, it can have a unique solution. Because there are more equations than​ variables, there are no free variables. If the system is consistent and there are no free​ variables, the solution set contains a unique solution. If the system is​ inconsistent, there is no solution.

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.

False. This statement is often​ true, but Span ​{u​,v​} is not a plane when v is a multiple of u or when u is the zero vector.

The set Span ​{u​,v​} is always visualized as a plane through the origin. Choose the correct answer below. True. The set Span ​{u​,v​} is always visualized as a plane in set of real numbers R^3 that contains u​, v​, and 0. True. The set Span ​{u​,v​} is always visualized as a line in set of real numbers R^3 that contains u​, v​, and 0. False. Although the set Span ​{u​,v​} is always visualized as a​ plane, it is not always through the origin. False. This statement is often​ true, but Span ​{u​,v​} is not a plane when v is a multiple of u or when u is the zero vector.

The row reduction algorithm applies only to augmented matrices for a linear system. Is this statement true or​ false? The statement is false. The algorithm applies to any​ matrix, whether or not the matrix is viewed as an augmented matrix for a linear system. The statement is true. Every matrix with at least two columns can be interpreted as the augmented matrix of a linear system. The statement is true. The row reduction algorithm is only useful when it is used to find the solution of a linear system. The statement is false. It is possible to create a linear system such that the row reduction algorithm does not apply to the corresponding augmented matrix.

The statement is false. The algorithm applies to any​ matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.

If one row in an echelon form of an augmented matrix is [ 0 0 0 5 0 ]​, then the associated linear system is inconsistent. Is this statement true or​ false? The statement is false. The indicated row corresponds to the equation 5x_4=​0, which means the system is consistent. The statement is true. The indicated row corresponds to the equation 5x_4=0. This equation is not a​ contradiction, so the linear system is inconsistent. The statement is true. The indicated row corresponds to the equation 5=0. This equation is a​ contradiction, so the linear system is inconsistent. The statement is false. The indicated row corresponds to the equation 5x_4=​0, which does not by itself make the system inconsistent.

The statement is false. The indicated row corresponds to the equation 5x_4=​0, which does not by itself make the system inconsistent.

Finding a parametric description of the solution set of a linear system is the same as solving the system. Is this statement true or​ false? The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has no more than one solution. The statement is true. Solving a linear system is the same as finding the solution set of the system. The solution set of a linear system can always be expressed using a parametric description. The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution. The statement is true. Regardless of whether a linear system has free​ variables, the solution set of the system can be expressed using a parametric description.

The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution.

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. Is this statement true or​ false? The statement is true. If a linear system has both basic and free​ variables, then each basic variable can be expressed in terms of the free variables. The statement is false. Not every linear system has basic variables. The statement is false. A variable that corresponds to a pivot column in the coefficient matrix is called a free​ variable, not a basic variable. The statement is true. It is the definition of a basic variable.

The statement is true. It is the definition of a basic variable.

Is the statement​ "A consistent system of linear equations has one or more​ solutions" true or​ false? Explain. ​True, a consistent system is defined as a system that has at least one solution. ​True, because a consistent system is made up of equations for planes in​ three-dimensional space. ​False, because a consistent system has infinitely many solutions. ​False, because a consistent system has only one unique solution.

True, a consistent system is defined as a system that has at least one solution.

Is the statement​ "Elementary row operations on an augmented matrix never change the solution set of the associated linear​ system" true or​ false? Explain. ​False, because the elementary row operations make a system inconsistent. ​False, because the elementary row operations augment the number of rows and columns of a matrix. ​True, because elementary row operations are always applied to an augmented matrix after the solution has been found. True, because the elementary row operations replace a system with an equivalent system.

True, because the elementary row operations replace a system with an equivalent system.

Is the statement​ "Two fundamental questions about a linear system involve existence and​ uniqueness" true or​ false? Explain. ​False, because two fundamental questions address whether it is possible to solve the system with row operations or whether a computer is necessary. ​False, because two fundamental questions address the type of row operations that can be used on the system and whether the linear operations fundamentally change the system. ​True, because two fundamental questions address whether the solution exists and whether there is only one solution. ​True, because two fundamental questions address whether the equations of the linear system exist in​ n-dimensional space and whether they can exist in more than one instance of​ n-dimensional space.

True, because two fundamental questions address whether the solution exists and whether there is only one solution.

If the columns of an m x n matrix A span set of real numbers R^m, then the equation Ax=b is consistent for each b in set of real numbers R^m. Choose the correct answer below. True. If the columns of A span set of real numbers R^m, then the equation Ax=b has a solution for each b in set of real numbers R^m. False. Since the columns of A span set of real numbers R^m, the matrix A has a pivot position in exactly m-1 rows. True. Since the columns of A span set of real numbers R^m, the augmented matrix [ A b ] has a pivot position in every row. False. If the columns of A span set of real numbers R^m, then the equation Ax=b is inconsistent for each b in set of real numbers R^m.

True. If the columns of A span set of real numbers R^m, then the equation Ax=b has a solution for each b in set of real numbers R^m.

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 equation Ax=b is unrelated to whether the vector b is a linear combination of the columns of a matrix A. False. If the matrix A is the identity​ matrix, then the equation Ax=b has at least one​ solution, but b is not a linear combination of the columns of A. False. If the equation Ax=b has infinitely many​ solutions, then the vector b cannot be a linear combination of the columns of A. True. The equation Ax=b has the same solution set as the equation x_1*a_1 + x_2*a_2 + ... + x_n*a_n = b

True. The equation Ax=b has the same solution set as the equation x_1*a_1 + x_2*a_2 + ... + x_n*a_n = b

The first entry in the product Ax is a sum of products. Choose the correct answer below. False. The first entry in Ax is the sum of the corresponding entries in x and the first entry in each column of A. True. The first entry in Ax is the sum of the products of corresponding entries in x and the first column of A. False. The first entry in Ax is the product of x_1 and the column a_1. 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.

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.

Two vectors are linearly dependent if and only if they lie on a line through the origin. Choose the correct answer below. True. Linearly dependent vectors must always intersect at the origin. True. Two vectors are linearly dependent if one of the vectors is a multiple of the other. Two such vectors will lie on the same line through the origin. False. Two vectors are linearly dependent if one of the vectors is a multiple of the other. The larger vector will be further from the origin than the smaller vector. False. If two vectors are linearly dependent then the graph of one will be​ orthogonal, or​ perpendicular, to the other.

True. Two vectors are linearly dependent if one of the vectors is a multiple of the other. Two such vectors will lie on the same line through the origin.

Section 1.9

True/False type questions from homework over section 1.9

Section 2.1

True/False type questions from homework over section 2.1

Section 1.3

True/False type questions from section 1.3 Homework

Section 1.7: Linearly Dependent or Independent

True/False type questions from section 1.7 homework

Section 1.5

True/false type questions from section 1.5 homework

Parametric vector form meaning

We call x = p + tv the equation of the line through p parallel to v. Thus, the solution set of Ax = b is a line through p parallel to the solution set of Ax = 0

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. False. ​Span{u​,v​} includes linear combinations of both u and v. False. ​Span{u​,v​} will not contain the origin. True. ​Span{u​,v​} is the set of all scalar multiples of u and all scalar multiples of v.

(a) No (b) Yes

answers on reverse side

Is the statement​ "The solution set of a linear system involving variables x_1, ..., x_n is a list of numbers (s_1, ..., s_n) that makes each equation in the system a true statement when the values s_1, ..., s_n are substituted for x_1, ..., x_n respectively" true or​ false? Explain. ​True, because the list of variables (x_1, ..., x_n) and the list of numbers (s_1 , ..., s_n) have a​ one-to-one correspondence. ​True, because the solution set of a linear system will have the same number of elements as the list of the variables in the system. ​False, because the list of numbers (s_1, ..., s_n) is the solution set for a linear system involving the variables x_1, ..., x_(n-1) . ​False, because the description applies to a single solution. The solution set consists of all possible solutions.

​False, because the description applies to a single solution. The solution set consists of all possible solutions.

Is the statement​ "Two equivalent linear systems can have different solution​ sets" true or​ false? Explain. ​False, because two systems are called equivalent only if they both have no solution. ​False, because two systems are called equivalent if they have the same solution set. ​True, because equivalent linear systems are systems with the same number of​ variables, which means that they can have different solution sets. ​True, because equivalent linear systems are systems that have the same number of rows and columns when they are written as augmented​ matrices, which means that they can have different solution sets.

​False, because two systems are called equivalent if they have the same solution set.

Is the statement​ "Every elementary row operation is​ reversible" true or​ false? Explain. False, because only interchanging is a reversible row operation. ​True, because interchanging can be reversed by​ scaling, and scaling can be reversed by replacement. ​True, because​ replacement, interchanging, and scaling are all reversible. ​False, because only scaling and interchanging are reversible row operations.

​True, because​ replacement, interchanging, and scaling are all reversible.