Math4A

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 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 more equations than unknowns is sometimes called an overdetermined system. Can such a system be​ consistent? Illustrate your answer with a specific system of three equations in two unknowns.

​​Yes, overdetermined systems can be consistent. For​ example, the system of equations below is consistent because it has the solution (2,4,24) left parenthesis 2 comma 4 right parenthesis. ​(Type an ordered​ pair.) x_1=2 , x_ 2 = 4 x_1 + x_2= 6

Suppose a system of linear equations has a 3x5 augmented matrix whose fifth column is not a pivot column. Is the system​ consistent? Why or why​ not? In the augmented matrix described​ above, is the rightmost column a pivot​ column?

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. That​ is, if and only if an echelon form of the augmented matrix has no row of the form​ [0 ... 0​ b] with b nonzero. NO

The equation Axequals0 gives an explicit description of its solution set. A. False. The equation Axequals0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set. B. True. Since the equation is​ solved, Axequals0 gives an explicit description of the solution set. C. True. The equation Axequals0 gives an explicit description of its solution set. Solving the equation amounts to finding an implicit description of its solution set. D. False. Since the equation is​ solved, Axequals0 gives an implicit description of its solution set.

A. False. The equation Axequals0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set.

The equation xequalspplustv describes a line through v parallel to p. A. False. The effect of adding p to v is to move v in a direction parallel to the plane through p and 0. So the equation xequalspplustv describes a plane through p parallel to v. B. False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation xequalspplustv describes a line through p parallel to v. C. False. The effect of adding p to v is to move p in a direction parallel to the plane through v and 0. So the equation xequalspplustv describes a plane through v parallel to p. D. True. The effect of adding p to v is to move p in a direction parallel to the line through v and 0. So the equation xequalspplustv describes a line through v parallel to p.

B. False. The effect of adding p to v is to move v in a direction parallel to the line through p and 0. So the equation xequalspplustv describes a line through p parallel to v.

The homogenous equation Axequals0 has the trivial solution if and only if the equation has at least one free variable. A. True. The homogeneous equation Axequals0 has the trivial solution if and only if the matrix A has a row of zeros which implies the equation has at least one free variable. B. False. The homogeneous equation Axequals0 always has the trivial solution. C. False. The homogeneous equation Axequals0 never has the trivial solution. D. True. The homogenous equation Axequals0 has the trivial solution if and only if the equation has at least one free variable which implies that the equation has a nontrivial solution.

B. False. The homogeneous equation Axequals0 always has the trivial solution. Your answer is correct.

The solution set of Axequalsb is the set of all vectors of the form wequalspplusBold v Subscript h​, where Bold v Subscript h is any solution of the equation Axequals0. A. False. The solution set could be empty. The statement is only true when the equation Upper A Bold x equals Bold b is inconsistent for some given b​, and there exists a vector p such that p is a solution. B. False. The solution set could be empty. The statement is only true when the equation Upper A Bold x equals Bold b is consistent for some given b​, and there exists a vector p such that p is a solution. C. False. The solution set could be the trivial solution. The statement is only true when the equation Upper A Bold x equals Bold b is inconsistent for some given b​, and there exists a vector p such that p is a solution. D. True. The equation Axequalsb is always consistent and there always exists a vector p that is a solution.

B. False. The solution set could be empty. The statement is only true when the equation Upper A Bold x equals Bold b is consistent for some given b​, and there exists a vector p such that p is a solution.

Every linear transformation is a matrix transformation. t/f

False. A matrix transformation is a special linear transformation of the form Fa x maps to Ax where A is a matrix.

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

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

The columns of a matrix A are linearly independent if the equation Axequals0 has the trivial solution. T/F

False. For every matrixFa ​ A, Axequals0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution

If S is a linearly dependent​ set, then each vector is a linear combination of the other vectors in S. T/F

False. If an indexed set of​ vectors, S, is linearly​ dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set.+

The weights c 1 comma ... comma c Subscript p in a linear combination c 1 Bold v 1 plus ... plus c Subscript p Baseline Bold v Subscript p cannot all be zero. true or false

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

If A is a 3 times 5 matrix and T is a transformation defined by ​T(x​)equalsAx​, then the domain of T is set of real numbers R ^3. t/f

False. The domain is actually Fa set of real numbers R Superscript 5​, because in the product Ax​, if A is an m times n matrix then x must be a vector in set of real numbers R Superscript n.

The equation Ax=b is referred to as a vector equation. True or false

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

The equation Ax=b is consistent if the augmented matrix [ab] has a pivot position in every row. True or false

False. The equation Axequalsb is referred to as a matrix equation because A is a matrix. If the augmented matrix [Ab] has a pivot position in every​ row, the equation Ax=b may or may not be consistent. One pivot position may be in the column representing b.

If A is an m times n ​matrix, then the range of the transformation x maps to Ax is set of real numbers R Superscript m. t/f

False. The range of the transformation is the set of all linear combinations of the columns of ​ A, because each image of the transformation is of the form Ax.

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.

In the echelon form of the augmented​ matrix, is there a row of the form​ [0 0 0 0​ b] with b​ nonzero? ​ Therefore, by the Existence and Uniqueness​ Theorem, the linear system is

NO consistent

A is a 3 times 3 matrix with three pivot positions. ​(a) Does the equation Axequals0 have a nontrivial​ solution? ​(b) Does the equation Axequalsb have at least one solution for every possible b​? ​(a) Does the equation Axequals0 have a nontrivial​ solution? No Yes ​(b) Does the equation Axequalsb have at least one solution for every possible b​? Yes No

No, Yes

Suppose A is a 4 times 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 for a_m x_m = b_m

Suppose A is a 5 times 7 matrix. How many pivot columns must A have if its columns span set of real numbers R^5​? ​Why?

The matrix must have Th 5 5 pivot columns. The statements​ "A has a pivot position in every​ row" and​ "the columns of A span set of real numbers R^5​" are logically equivalent.

Suppose A is a 7 times 5 matrix. How many pivot columns must A have if its columns are linearly​ independent? Why?

The matrix must have Th 5 5 pivot columns.​ Otherwise, the equation Ax= 0 would have a free​ variable, in which case the columns of A would be linearly dependent.

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. 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. 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.

If one row in an echelon form of an [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 does not by itself make the system inconsistent. The equation 5x_4 = 0 yields x_4=​0, which is not a contradiction. The associated linear system may be consistent or​ inconsistent, depending on the other rows of the echelon form of the augmented matrix.

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 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. It is the definition of a basic variable.

The columns of any 4 times 5 matrix are linearly dependent. T/F

True. A 4 times 5 matrix has more columns than​ rows, and if a set contains more vectors than there are entries in each​ vector, then the set is linearly dependent.

A linear transformation is a special type of function. T/F

True. A linear transformation is a function from Tr set of real numbers R Superscript n to set of real numbers R Superscript m that assigns to each vector x in set of real numbers R Superscript n a vector ​T(x​) in set of real numbers R Superscript m.

Asking whether the linear system corresponding to an augmented matrix left bracket Start 1 By 4 Matrix 1st Row 1st Column Bold a 1 2nd Column Bold a 2 3rd Column Bold a 3 4st Column Bold b EndMatrix right bracket has a solution amounts to asking whether b is in Span StartSet Bold a 1 comma Bold a 2 comma Bold a 3 EndSet.

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.

If A is an m times n matrix and if the equation Ax=b is inconsistent for some b in set of real numbers R Superscript m​, then A cannot have a pivot position in every row. T/F

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

If the columns of an m times n matrix A span set of real numbers R Superscript m​, then the equation Ax=b is consistent for each b in set of real numbers R Superscript m. Choose the correct answer below. True or false

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

If x and y are linearly​ independent, and if ​{x​, y​, z​} is linearly​ dependent, then z is in ​Span{x​, y​}. Choose the correct answer below. T/F

True. If ​{x​, y​, z​} is linearly​ dependent, then z must be a linear combination of x and y because x and y are linearly independent. So z is in ​Span{x​, y​}.

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 True or false

True. The equation Axequalsb 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 b.

The first entry in the product Ax is a sum of products. Choose the correct answer below. True of false

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.

A transformation T is linear if and only if Tleft parenthesis c 1 Bold v 1 plus c 2 Bold v 2 right parenthesisequalsc 1 Upper T left parenthesis Bold v 1 right parenthesis plus c 2 Upper T left parenthesis Bold v 2 right parenthesis for all Bold v 1 and Bold v 2 in the domain of T and for all scalars c 1 and c 2. t/f

True. This equation correctly summarizes the properties necessary for a transformation to be linear.Tr

Any list of five real numbers is a vector in set of real numbers R⁵

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

A is a 2 times 5 matrix with two pivot positions. ​(a) Does the equation Axequals0 have a nontrivial​ solution? ​(b) Does the equation Axequalsb have at least one solution for every possible b​? ​(a) Does the equation Axequals0 have a nontrivial​ solution? No Yes ​(b) Does the equation Axequalsb have at least one solution for every possible b​? Yes No

Yes NO

Suppose Upper A Bold x equals Bold b has a solution. Explain why the solution is unique precisely when Upper A Bold x equals Bold 0 has only the trivial solution. Choose the correct answer. A. Since Upper A Bold x equals Bold b is​ consistent, then the solution is unique if and only if there is at least one free variable in the corresponding system of equations. This happens if and only if the equation Upper A Bold x equals Bold 0 has only the trivial solution. B. Since Upper A Bold x equals Bold b is​ consistent, its solution set is obtained by translating the solution set of Upper A Bold x equals Bold 0. So the solution set of Upper A Bold x equals Bold b is a single vector if and only if the solution set of Upper A Bold x equals Bold 0 is a single​ vector, and that happens if and only if Upper A Bold x equals Bold 0 has only the trivial solution. C. Since Upper A Bold x equals Bold b is​ inconsistent, its solution set is obtained by translating the solution set of Upper A Bold x equals Bold 0. For Upper A Bold x equals Bold b to be​ inconsistent, Upper A Bold x equals Bold 0 has only the trivial solution. D. Since Upper A Bold x equals Bold b is​ inconsistent, then the solution set of Upper A Bold x equals Bold 0 is also inconsistent. The solution set of Upper A Bold x equals Bold 0 is inconsistent if and only if Upper A Bold x equals Bold 0 has only the trivial solution.

B. Since Upper A Bold x equals Bold b is​ consistent, its solution set is obtained by translating the solution set of Upper A Bold x equals Bold 0. So the solution set of Upper A Bold x equals Bold b is a single vector if and only if the solution set of Upper A Bold x equals Bold 0 is a single​ vector, and that happens if and only if Upper A Bold x equals Bold 0 has only the trivial solution.

A homogeneous equation is always consistent. A. True. A homogenous equation can be written in the form Axequals0​, where A is an m times n matrix and 0 is the zero vector in set of real numbers R Superscript m. Such a system Axequals0 always has at least one nontrivial solution. Thus a homogenous equation is always consistent. B. False. A homogenous equation can be written in the form Axequals0​, where A is an m times n matrix and 0 is the zero vector in set of real numbers R Superscript m. Such a system Axequals0 always has at least one​ solution, namely, xequals0. Thus a homogenous equation is always inconsistent. C. True. A homogenous equation can be written in the form Axequals0​, where A is an m times n matrix and 0 is the zero vector in set of real numbers R Superscript m. Such a system Axequals0 always has at least one​ solution, namely, xequals0. Thus a homogenous equation is always consistent. D. False. A homogenous equation can be written in the form Axequals0​, where A is an m times n matrix and 0 is the zero vector in set of real numbers R Superscript m. Such a system Axequals0 always has at least one nontrivial solution. Thus a homogenous equation is always inconsistent.

C. True. A homogenous equation can be written in the form Axequals0​, where A is an m times n matrix and 0 is the zero vector in set of real numbers R Superscript m. Such a system Axequals0 always has at least one​ solution, namely, xequals0. Thus a homogenous equation is always consistent. Your answer is correct.