Test 1
A linear system is consistent if and only if the rightmost column of the augmented matrix_____________a pivot column. That is, if and only if an echelon form of the augmented matrix has_____________ of the form [0 ... 0 b] with b nonzero.
is not; now row
The vector v results when a vector u-v is added to the vector v.
False. Adding u - v to v results in u.
An example of a linear combination of vectors v1 and v2 is the vector (1/2) v1. Choose the correct answer below.
True, as (1/2)v1 = (1/2) v1 plus 0v2.
The first entry in the product Ax is a sum of products. 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 points in the plane corresponding to [-2; 5] and [-5; 2] lie on a line through the origin. Choose the correct answer below.
False if [-2; 5] and [-5; 2] were on the line through the origin, then [-5,2] would be a multiple of [-2;5].
Is the statement "A 5 times 6 matrix has six rows" true or false? Explain.
False, because a 5 times 6 matrix has five rows and six columns.
f the coefficient matrix A has a pivot position in every row, then the equation Ax = b is inconsistent. 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 Ax = b must be consistent.
The equation Ax = b is consistent if the augmented matrix left bracket [A b] has a pivot position in every row. Choose the correct answer below.
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.
If A is an m * n matrix whose columns do not span set of real numbers R^m, then the equation Ax = b is consistent for every b in set of real numbers R Superscript m. Choose the correct answer below.
False. If the columns of A do not span set of real numbers R^m, then A does not have a pivot position in every row, and row reducing [A b] could result in a row of the form [0 0 .... 0 c] , where c is a nonzero real number.
The weights c1.....cp in a linear combination c 1*v 1 + ... + cp*vp cannot all be zero.
False. Setting all the weights equal to zero results in the vector 0.
Another notation for the vector [ 4; 3] is [4, 3] Choose the correct answer below.
False. The alternative notation for a (column) vector is (-4,3), using parentheses and commas.
he 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 matrix equation because A is a matrix.
The set Span {u,v} is always visualized as a plane through the origin. Choose the correct answer below.
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.
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.
Suppose A is a 4 x 3 matrix and b is a vector in set of real numbers R^4 with the property that Ax = b has a unique solution. What can you say about the reduced echelon form of A? Justify your answer.
The first 3 rows will have a pivot position and the last row will be all zeros. If a row had more than one 1, then there would be an infinite number of solutions am + xm = b.
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
The row reduction algorithm applies only to augmented matrices for a linear system.
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 echelon form of a matrix is unique. Choose the correct answer below.
The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.
Whenever a system has free variables, the solution set contains many solutions. Choose the correct answer below.
The statement is false. The existence of at least one solution is not related to the presence or absence of free variables. If the system is inconsistent, the solution set is empty.
If one row in an echelon form of an augmented matrix is left bracket Start 1 By 5 Matrix 1st Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 5 5st Column 0 EndMatrix right bracket , then the associated linear system is inconsistent.
The statement is false. The indicated row corresponds to the equation 5x 4equals0, which does not by itself make the system inconsistent.
The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process. Choose the correct answer below.
The statement is false. The pivot positions in a matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix.
Finding a parametric description of the solution set of a linear system is the same as solving the system.
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.
The statement is true. It is the definition of a basic variable.
Reducing a matrix to echelon form is called the forward phase of the row reduction process. Choose the correct answer below.
The statement is true. Reducing a matrix to echelon form is called the forward phase and reducing a matrix to reduced echelon form is called the backward phase.
A general solution of a system is an explicit description of all solutions of the system. Choose the correct answer below.
The statement is true. The row reduction algorithm leads directly to an explicit 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.
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
Asking whether the linear system corresponding to an augmented matrix left bracket [a1, a2, a3, b]has a solution amounts to asking whether b is in Span {a1,a2,a3}
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.
A is an m * 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.
True. If A is an m * 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.
The solution set of a linear system whose augmented matrix is left bracket [a1, a2, a3, b] is the same as the solution set of Ax = b, if A= [a1, a2, a3] . Choose the correct answer below.
True. If A is an m times n matrix with columns [a1,a2,... an] and b is a vector in set of real numbers R^m, the matrix equation Ax = b has the same solution set as the system of linear equations whose augmented matrix [a1, a2, .... an b];
If the columns of an m * n matrix A span set of real numbers Rm, then the equation Ax = b is consistent for each b in set of real numbers Rm. 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.
The solution set of the linear system whose augmented matrix is [a1, a2, a3, b] is the same as the solution set of the equation x1*a1+x2*a2+x3*a3 = b. Choose the correct answer below.
True. The augmented matrix for x1*a1+x2*a2+x3*a3 = b is [a1, a2, a3, b]
If the equation Ax = b is consistent, then b is in the set spanned by the columns of A. Choose the correct answer below.
True. The equation Ax = b has a nonempty solution set if and only if b is a linear combination of the columns of A.
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 has the same solution set as the equation x 1*a 1 + x 2 * a 2 +x n *a n = b.
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.
True. The matrix A is the matrix of coefficients of the system of vectors.
Every matrix equation Ax = b corresponds to a vector equation with the same solution set. Choose the correct answer below.
True. The matrix equation Ax=b is simply another notation for the vector equation x1 *a1 + x2 *a2 +... + xn * an, are the columns of A.
b. Any list of five real numbers is a vector in R^5
True. set of real numbers R^5 denotes the collection of all lists of five real numbers.
Let A be a 3 x 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in set of real numbers R^3. Generalize your argument to the case of an arbitrary A with more rows than columns. Why is the equation Ax = b not consistent for all b in set of real numbers R^3?
When written in reduced row echelon form, any 3 times 2 matrix will have at least one row of all zeros. When solving Ax = b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
Let A be a 3 x 2 matrix. Explain why the equation Ax = b cannot be consistent for all b in set of real numbers R^3. Generalize your argument to the case of an arbitrary A with more rows than columns. Let A be an m times n matrix, where m > n. Why is Ax = b not consistent for all b in set of real numbers R Superscript m?
When written in reduced row echelon form, any m times n matrix will have at least one row of all zeros. When solving Ax = b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
Is the statement "Two matrices are row equivalent if they have the same number of rows" true or false? Explain.
False, because if two matrices are row equivalent it means that there exists a sequence of row operations that transforms one matrix to the other.
Is the statement "The solution set of a linear system involving variables x 1 ........x n (s1, ... sn) that makes each equation in the system a true statement when the values s1...sn are substituted for x 1...xn Baseline comma respectively" true or false? Explain.
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 if they have the same solution set.
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.
Is the statement "Elementary row operations on an augmented matrix never change the solution set of the associated linear system" true or false? Explain.
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.
True, because two fundamental questions address whether the solution exists and whether there is only one solution.
Is the statement "Every elementary row operation is reversible" true or false? Explain.
True, because replacement, interchanging, and scaling are all reversible.