Linear Algebra True/False Exam 1
Suppose F is a 5 times 55×5 matrix whose column space is not equal to set of real numbers R Superscript 5ℝ5. What can you say about Nul F?
If Col Upper F not equals set of real numbers R Superscript 5F≠ℝ5, then the columns of F do not span set of real numbers R Superscript 5ℝ5. Since F is square, the Invertible Matrix Theorem shows that F is not invertible and the equation Fxequals=0 has a nontrivial solution. Therefore, Nul F contains a nonzero vector.
If Q is a 4 times ×4 matrix and Col Qequals=set of real numbers R Superscript 4ℝ4, what can you say about solutions of equations of the form Qxequals=b for b in set of real numbers R Superscript 4ℝ4?
If Col Qequals=set of real numbers R Superscript 4ℝ4, then the columns of Q span set of real numbers R Superscript 4ℝ4. Since Q is square, the Invertible Matrix Theorem shows that Q is invertible and the equation Qxequals=b has a solution for each b in set of real numbers R Superscript 4ℝ4.
Suppose A is n times ×n and the equation Axequals=b has a solution for each b in set of real numbers R Superscript nℝn. Explain why A must be invertible. [Hint: Is A row equivalent to Upper I Subscript nIn?]
If the equation Axequals=b has a solution for each b in set of real numbers R Superscript nℝn, then A has a pivot position in each row. Since A is square, the pivots must be on the diagonal of A. It follows that A is row equivalent to Upper I Subscript nIn. Therefore, A is invertible.
Let A be a 3 times ×3 matrix with two pivot positions. Use this information to answer parts (a) and (b) below. b. Does the equation Upper A Bold x equals Bold bAx=b have at least one solution for every possible b?
No. A has one free variable. To have at least one solution for every possible b, A cannot have any free variable.
Suppose Upper A Bold x equals Bold bAx=b has a solution. Explain why the solution is unique precisely when Upper A Bold x equals Bold 0Ax=0 has only the trivial solution.
Since Upper A Bold x equals Bold bAx=b is consistent, its solution set is obtained by translating the solution set of Upper A Bold x equals Bold 0Ax=0. So the solution set of Upper A Bold x equals Bold bAx=b is a single vector if and only if the solution set of Upper A Bold x equals Bold 0Ax=0 is a single vector, and that happens if and only if Upper A Bold x equals Bold 0Ax=0 has only the trivial solution.
Why is the statement above true?
The columns of a matrix A are linearly independent if and only if Axequals=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 colunm.
How many rows and columns must a matrix A have in order to define a mapping from set of real numbers R Superscript 5ℝ5 into set of real numbers R Superscript 6ℝ6 by the rule T(x)equals=Ax?
The matrix A must have 6 rows and 5 columns.
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 5 pivot columns. Otherwise, the equation ABold xxequals=Bold 00 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.
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.
If one row in an echelon form of an augmented matrix is [ 0 0 0 5 0 ], then the associated linear system is inconsistent.
The statement is false. The indicated row corresponds to the equation 5x 4x4equals=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.
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.
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 "Every elementary row operation is reversible" true or false? Explain.
True, because replacement, interchanging, and scaling are all reversible.
In order for a matrix B to be the inverse of A, both equations ABequals=I and BAequals=I must be true.
True, by definition of invertible.
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.
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}.
The columns of any 4 times ×5 matrix are linearly dependent. Choose the correct answer below.
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 homogeneous equation is always consistent.
True. A homogenous equation can be written in the form Axequals=0, where A is an m times ×n matrix and 0 is the zero vector in set of real numbers R Superscript mℝm. Such a system Axequals=0 always has at least one solution, namely, xequals=0. Thus a homogenous equation is always consistent.
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. 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 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 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.
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 comma ... comma x Subscript nx1, ..., xn is a list of numbers left parenthesis s 1 comma ... comma s Subscript n Baseline right parenthesiss1, ..., sn that makes each equation in the system a true statement when the values s 1 comma ... comma s Subscript ns1, ..., sn are substituted for x 1 comma ... comma x Subscript n Baseline commax1, ..., xn, 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.
If A and B are n times nn×n and invertible, then Upper A Superscript negative 1 Baseline Upper B Superscript negative 1A−1B−1 is the inverse of AB.
False; Upper B Superscript negative 1 Baseline Upper A Superscript negative 1B−1A−1 is the inverse of AB.
If A is an n times nn×n matrix, then the equation Axequals=b has at least one solution for each b in set of real numbers R Superscript nℝn.
False; by the Invertible Matrix Theorem Axequals=b has at least one solution for each b in set of real numbers R Superscript nℝn only if a matrix is invertible.
If Aequals=Start 2 By 2 Table 1st Row 1st Column a 2nd Column b 2nd Row 1st Column c 2nd Column d EndTableabcd and abminus−cdnot equals≠0, then A is invertible.
False; if adminus−bcnot equals≠0, then A is invertible.
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 "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.
If the equation Axequals=0 has a nontrivial solution, then A has fewer than n pivot positions.
True; by the Invertible Matrix Theorem if the equation Axequals=0 has a nontrivial solution, then matrix A is not invertible. Therefore, A has fewer than n pivot positions.
If the equation Axequals=0 has only the trivial solution, then A is row equivalent to the n times nn×n identity matrix.
True; by the Invertible Matrix Theorem if the equation Axequals=0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the n times nn×n identity matrix.
Each elementary matrix is invertible
True; since each elementary matrix corresponds to a row operation, and every row operation is reversible, every elementary matrix has an inverse matrix.
Let A be a 5 times ×4 matrix. What must a and b be in order to define Upper T : set of real numbers R Superscript a Baseline right arrow set of real numbers R Superscript bT : ℝa→ℝb by T(x)equals=Ax?
a = 4 b = 5
If A is an m times ×n matrix, then the columns of A are linearly independent if and only if A has _______ pivot columns.
n
The equation xequals=pplus+tv describes a line through v parallel to p.
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 xequals=pplus+tv describes a line through p parallel to v.
The equation Axequals=0 gives an explicit description of its solution set.
False. The equation Axequals=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set.
The equation Axequals=b is referred to as a vector equation. Choose the correct answer below.
False. The equation Axequals=b is referred to as a matrix equation because A is a matrix.
The homogenous equation Axequals=0 has the trivial solution if and only if the equation has at least one free variable.
False. The homogeneous equation Axequals=0 always has the trivial solution.
The solution set of Axequals=b is the set of all vectors of the form wequals=pplus+Bold v Subscript hvh, where Bold v Subscript hvh is any solution of the equation Axequals=0.
False. The solution set could be empty. The statement is only true when the equation Upper A Bold x equals Bold bAx=b is consistent for some given b, and there exists a vector p such that p is a solution.
A is a 2 times ×5 matrix with two pivot positions. Does the equation Axequals=0 have a nontrivial solution?
yes
A is a 2 times ×5 matrix with two pivot positions. Does the equation Axequals=b have at least one solution for every possible b?
yes
The standard matrix of a horizontal shear transformation from set of real numbers R squaredℝ2 to set of real numbers R squaredℝ2 has the form left bracket Start 2 By 2 Matrix 1st Row 1st Column a 2nd Column 0 2nd Row 1st Column 0 2nd Column d EndMatrix right bracketa00d. Choose the correct answer below.
False. The standard matrix has the form left bracket Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column k 2nd Row 1st Column 0 2nd Column 1 EndMatrix right bracket1k01.
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. Choose the correct answer below.
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.
Explain why the columns of an n×n matrix A are linearly independent when A is invertible.
If A is invertible, then the equation ABold xxequals=Bold 00 has the unique solution Bold xxequals=Bold 00. Since ABold xxequals=Bold 00 has only the trivial solution, the columns of A must be linearly independent.
If a matrix A is 44 times ×88 and the product AB is 44 times ×55, what is the size of B?
B is an 8 by 5 matirx
The equation Axequals=b is consistent if the augmented matrix left bracket Start 1 By 2 Matrix 1st Row 1st Column Upper A 2nd Column Bold b EndMatrix right bracketAb has a pivot position in every row. Choose the correct answer below.
False. If the augmented matrix left bracket Start 1 By 2 Matrix 1st Row 1st Column Upper A 2nd Column Bold b EndMatrix right bracketAb has a pivot position in every row, the equation equation Axequals=b may or may not be consistent. One pivot position may be in the column representing b.
If A is a 4 times ×3 matrix, then the transformation x maps to↦ Ax maps set of real numbers R cubedℝ3 onto set of real numbers R Superscript 4ℝ4. Choose the correct answer below.
False. The columns of A do not span set of real numbers R Superscript 4ℝ4.
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.
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 "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.
A mapping T: set of real numbers R Superscript nℝnmaps to↦set of real numbers R Superscript mℝm is one-to-one if each vector in set of real numbers R Superscript nℝn maps onto a unique vector in set of real numbers R Superscript mℝm. Choose the correct answer below.
False. A mapping T is said to be one-to-one if each b in set of real numbers R Superscript mℝm is the image of at most one x in set of real numbers R Superscript nℝn.
A mapping T: set of real numbers RℝSuperscript nnright arrow→set of real numbers RℝSuperscript mm is onto if every vector x in set of real numbers RℝSuperscript nn maps onto some vector in set of real numbers RℝSuperscript mm. Choose the correct answer below.
False. A mapping T: set of real numbers RℝSuperscript nnright arrow→set of real numbers RℝSuperscript mm is onto if every vector in set of real numbers RℝSuperscript mm is mapped onto by some vector x in set of real numbers RℝSuperscript nn.
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. Choose the correct answer below.
False. A transformation is linear if T(uplus+v)equals=T(u)plus+T(v) and T(cu)equals=cT(u) for all vectors u, v, and scalars c. The first transformation results in some vector u, so the properties of a linear transformation must still apply when two transformations are applied.
If A is a 3 times ×2 matrix, then the transformation xmaps to↦Ax cannot be one-to-one. Choose the correct answer below.
False. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If Axequals=b does not have a free variable, then the transformation represented by A is one-to-one.
The columns of a matrix A are linearly independent if the equation Axequals=0 has the trivial solution. Choose the correct answer below.
False. For every matrix A, Axequals=0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution.
A vector b is a linear combination of the columns of a matrix A if and only if the equation Axequals=b has at least one solution. Choose the correct answer below.
True. The equation Axequals=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.
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 columns of the standard matrix for a linear transformation from set of real numbers R Superscript nℝn to set of real numbers R Superscript mℝm are the images of the columns of the n times ×n identity matrix under T. Choose the correct answer below.
True. The standard matrix is the m times ×n matrix whose jth column is the vector Upper T left parenthesis Bold e Subscript j right parenthesisTej, where Bold e Subscript jej is the jth column of the identity matrix in set of real numbers R Superscript nℝn.
If T: set of real numbers Rℝsquared2right arrow→set of real numbers Rℝsquared2 rotates vectors about the origin through an angle phiφ, then T is a linear transformation. Choose the correct answer below.
True. The standard matrix, A, of the linear transformation is Start 2 By 2 Table 1st Row 1st Column cosine phi 2nd Column negative sine phi 2nd Row 1st Column sine phi 2nd Column cosine phi EndTablecosφ−sinφsinφcosφ.
A linear transformation T: set of real numbers RℝSuperscript nnright arrow→set of real numbers RℝSuperscript mm is completely determined by its effect on the columns of the n times ×n identity matrix. Choose the correct answer below.
True. The vector x can be written as a linear combination of the columns of the identity matrix. T is a linear transformation so T(x) can be written as a linear combination of the vectors T(Bold e 1e1) and T(Bold e 2e2).
Every linear transformation from set of real numbers R Superscript nℝn to set of real numbers R Superscript mℝm is a matrix transformation. Choose the correct answer below.
True. There exists a unique matrix A such that Upper T left parenthesis Bold x right parenthesis equals Upper A Bold xT(x)=Ax for all x in set of real numbers R Superscript nℝn.
If Upper A Superscript Upper TAT is not invertible, then A is not invertible.
True; by the Invertible Matrix Theorem if Upper A Superscript Upper TAT is not invertible all statements in the theorem are false, including A is invertible. Therefore, A is not invertible.
If A is an invertible n×n matrix, then the equation ABold xxequals=Bold bb is consistent for each Bold bb in set of real numbers R Superscript nℝn
True; since A is invertible, Upper A Superscript negative 1A−1Bold bb exists for all Bold bb in set of real numbers R Superscript nℝn. Define Bold xxequals=Upper A Superscript negative 1A−1Bold bb. Then ABold xxequals=Bold bb.
If the columns of A span set of real numbers R Superscript nℝn, then the columns are linearly independent.
True; the Invertible Matrix Theorem states that if the columns of A span set of real numbers R Superscript nℝn, then matrix A is invertible. Therefore, the columns are linearly independent.
Let A be a 3 times ×3 matrix with two pivot positions. Use this information to answer parts (a) and (b) below. a. Does the equation Upper A Bold x equals Bold 0Ax=0 have a nontrivial solution?
Yes. Since A has 2 pivots, there is one free variable. So Upper A Bold x equals Bold 0Ax=0 has a nontrivial solution.
