Linear Algebra (Exam 1)T/F
Let T: ℝn→ℝm be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in ℝn. Explain why the set {T(v1), T(v2), T(v3)} is linearly dependent.
Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1+c2v2+c3v3=0.It follows that c1T(v1)+c2T(v2)+c3 T(v3)=0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent.
Use matrix algebra to show that if A is invertible and D satisfies AD=I, then D=A−1.
Left-multiply each side of the equation AD=I by A−1 to obtain A−1AD=A−1I, ID=A−1, and D=A−1.
Suppose A is a 3×3 matrix and y is a vector in ℝ3 such that the equation Ax=y does not have a solution. Does there exist a vector z in ℝ3 such that the equation Ax=z has a unique solution?
No. Since Ax=y has no solution, then A cannot have a pivot in every row. So the equation Ax=z has at most two basic variables and at least one free variable for any z. Thus the solution set for Ax=z is either empty or has infinitely many elements.
Could a set of n vectors in ℝm span all of ℝm when n is less than m? 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).
Suppose Ax=b has a solution. Explain why the solution is unique precisely when Ax=0 has only the trivial solution.
Since Ax=b is consistent, its solution set is obtained by translating the solution set of Ax=0. So the solution set of Ax=b is a single vector if and only if the solution set of Ax=0 is a single vector, and that happens if and only if Ax=0 has only the trivial solution.
Suppose A is a 5×7 matrix. How many pivot columns must A have if its columns span ℝ5? Why?
The matrix must have 5 pivot columns. The statements "A has a pivot position in every row" and "the columns of A span ℝ5" are logically equivalent.
Suppose A is a 7×5 matrix. How many pivot columns must A have if its columns are linearly independent? Why?
The matrix must have 5pivot columns. Otherwise, the equation Ax=0would have a free variable, in which case the columns of A would be linearly dependent.
A 5×6 matrix has six rows.
The statement is false. A 5×6 matrix has five rows and six columns.
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. (AB)T=ATBT
The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or (AB)T=BTAT.
If v1 and v2 are in ℝ4 and v2 is not a scalar multiple of v1, then {v1,v2} is linearly independent.
The statement is false. The vector v could be the zero vector.
A general solution of a system is an explicit description of all solutions of the system.
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.
The columns of the standard matrix for a linear transformation from ℝn to ℝm are the images of the columns of the n×n identity matrix.
The statement is true. The standard matrix is the m×n matrix whose jth column is the vector Tej, where ej is the jth column of the identity matrix in ℝn.
If T: ℝ2→ℝ2 rotates vectors about the origin through an angle φ, then T is a linear transformation.
The statement is true. The standard matrix, A, of the linear transformation is [cosφ −sinφ][sinφ cosφ]
The effect of adding p to a vector is to move the vector in a direction parallel to p.
True. Given v and p in set of real numbers ℝ² or set of real numbers ℝ³, the effect of adding p to v is to move v in a direction parallel to the line through p and 0.
If A and B are n×n and invertible, then A−1B−1 is the inverse of AB.
The statement is false. The inverse of AB is B−1A−1.
A linear transformation preserves the operations of vector addition and scalar multiplication.
The statement is true. The linear transformation T(cu+dv) is the same as cT(u)+dT(v) in ℝm. Therefore, vector addition and scalar multiplication are preserved.
In order for a matrix B to be the inverse of A, both equations AB=I and BA=I must be true.
The statement is true. The product of a matrix and its inverse is the identity matrix.
Let A and B be arbitrary matrices for which the indicated sum is defined. Determine whether the statement below is true or false. Justify the answer. AT+BT= (A+B)T
The statement is true. The transpose property states that (A+B)T=AT+BT.
Two fundamental questions about a linear system involve existence and uniqueness.
The statement is true. The two fundamental questions are about whether the solution exists and whether there is only one solution.
A linear transformation T: ℝn→ℝm is completely determined by its effect on the columns of the n×n identity matrix.
The statement is 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(e1) through T(en).
If the columns of an m×n matrix A span ℝm, then the equation Ax=b is consistent for each b in ℝm.
his statement is true. If the columns of A span ℝm, then the equation Ax=b has a solution for each b in ℝm.
Suppose a system of linear equations has a 3×5 augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why or why not?
is not, no row, no, no, consistent
If A=abcd and ab−cd≠0, then A is invertible.
The statement is false. If ad−bc≠0, then A is invertible.
If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span{x, y}.
The statement is 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 4×5 matrix are linearly dependent.
The statement is true. A 4×5 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.
The transpose of a sum of matrices equals the sum of their transposes.
The statement is true. This is a generalized statement that follows from the theorem (A+B)T=AT+BT.
Suppose a x×y coefficient matrix for a system has x pivot columns. Is the system consistent? Why or why not?
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have y+1 columns and will not have a row of the form 000001, so the system is consistent
The equation Ax=b is referred to as a vector equation.
This statement is false. The equation Ax=b is referred to as a matrix equation because A is a matrix.
The solution set of a linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of Ax=b, if A= [a1 a2 a3].
This statement is true. If A is an m×n matrix with columns [a1 a2 ⋯an] and b is a vector in ℝm, the matrix equation Ax=b has the same solution set as the system of linear equations whose augmented matrix is [a1 a2 ⋯an b].
If v1, ..., v4 are in ℝ4 and v3=2v1+v2, then v1, v2, v3, v4 is linearly dependent.
True. The vector V3 is a linear combination of V1 and V2, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent.
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 b≠0, can the solution set of Ax=b be a plane through the origin?
No. If the solution set of Ax=b contained the origin, then 0 would satisfy A0=b, which is not true since b is not the zero vector.
Could a set of three vectors in ℝ4 span all of ℝ4? Explain.
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).
Let v1=[1 0 −1 0], v2= [0 −1 0 1], and v3=[ 0 0 −1 1]. Does v1,v2,v3 span ℝ4? Why or why not?
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only three rows.
Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you say about the columns of A?
The columns of A are linearly dependent because if the last column in B is denoted bp, then the last column of AB can be rewritten as Abp=0. Since bp is not all zeros, then any solution to Abp=0 can not be the trivial solution
Suppose A is a 4×3 matrix and b is a vector in ℝ4 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 am xm = bm.
Suppose the first two columns, b1 and b2, of B are equal. What can you say about the columns of AB (if AB is defined)? Why?
The first two columns of AB are Ab1 and Ab2. They are equal since b1 and b2 are equal.
Suppose A is a 3×3 matrix and b is a vector in ℝ3 with the property what Ax=b has a unique solution. Explain why the columns of A must span ℝ3.
The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation Ax=b. Therefore the columns of A must span ℝ3.
A mapping T: ℝn→ℝm is one-to-one if each vector in ℝn maps onto a unique vector in ℝm.
The statement is false. A mapping T is said to be one-to-one if each b in ℝm is the image of at most one x in ℝn
A mapping T: ℝn→ℝm is onto ℝm if every vector x in ℝn maps onto some vector in ℝm.
The statement is false. A mapping T: ℝn→ℝm is onto ℝm if every vector in ℝm is mapped onto by some vector x in ℝn.
Every linear transformation is a matrix transformation.
The statement is false. A matrix transformation is a special linear transformation of the form x↦Ax where A is a matrix.
If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.
The statement is false. A nontrivial solution of Ax=0is a nonzero vector x that satisfies Ax=0. Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
The statement is false. A transformation is linear if T(u+v)=T(u)+T(v) and T(cu)=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 3×2 matrix, then the transformation x↦Ax cannot be one-to-one.
The statement is 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 Ax=b does not have a free variable, then the transformation represented by A is one-to-one.
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 columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution.
The statement is false. For every matrix A, Ax=0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution.
A product of invertible n×n matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
The statement is false. If A and B are invertible matrices, then (AB)−1=B−1A−1.
he points in the plane corresponding to [−2] [ 5] and [−5] [2] lie on a line through the origin.
The statement is false. If [−2 5] and [−5 2] were on the line through the origin, then [−5 2] would be a multiple of [−2 5], which is not the case.
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
The statement is 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 set Span {u, v} is always visualized as a plane through the origin.
The statement is false. It 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 weights c1, ..., cp in a linear combination c1v1+⋯+cpvp cannot all be zero.
The statement is false. Setting all the weights equal to zero results in the vector 0.
Another notation for the vector [−4] [ 3 ] is [−4 3].
The statement is false. The alternative notation for a (column) vector is (−4,3), using parentheses and a comma.
Let A, B, and C be arbitrary matrices for which the indicated products are defined. Determine whether the statement below is true or false. Justify the answer. (AB)C=(AC)B
The statement is false. The associative law of multiplication for matrices states that A(BC)=(AB)C.
Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
The statement is false. The definition of AB states that each column of AB is a linear combination of the columns of A using weights from the corresponding column of B
If A is a 3×5 matrix and T is a transformation defined by T(x)=Ax,then the domain of T is ℝ3.
The statement is false. The domain is actually ℝ5, because in the product Ax, if A is an m×n matrix then x must be a vector in ℝn.
The echelon form of a matrix is unique
The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.
The equation x=p+tv describes a line through v parallel to p.
The statement is 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 x=p+tv describes a line through p parallel to v.
The equation Ax=0 gives an explicit description of its solution set.
The statement is false. The equation Ax=0 gives an implicit description of its solution set. Solving the equation amounts to finding an explicit description of its solution set.
The solution set of a linear system involving variables x1, ..., xn is a list of numbers s1, ..., sn that makes each equation in the system a true statement when the values s1, ..., sn are substituted for x1, ..., xn, respectively.
The statement is false. The given description is of a single solution of such a system. The solution set of the system consists of all possible solutions.
The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.
The statement is false. The homogeneous equation Ax=0 always has the trivial solution.
If A and B are 3×3 matrices and B=[b1 b2 b3], then AB=[Ab1+Ab2+Ab3].
The statement is false. The matrix [Ab1+Ab2+Ab3]is a 3×1 matrix, and AB must be a 3×3 matrix. The plus signs should be spaces between the 3 columns.
The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
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.
If T: ℝn → ℝm is a linear transformation and if c is in ℝm, then a uniqueness question is "Is c in the range of T?"
The statement is false. The question "is c in the range of T?" is the same as "does there exist an x whose image is c?" This is an existence question.
If A is an m×n matrix, then the range of the transformation x↦Ax is ℝm.
The statement is 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.
The solution set of Ax=b is the set of all vectors of the form w=p +Vh, where Vh is any solution of the equation Ax=0.
The statement is false. The solution set could be empty. The statement is only true when the equation Ax=b is consistent for some given b, and there exists a vector p such that p is a solution.
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.
The transpose of a product of matrices equals the product of their transposes in the same order.
The statement is false. The transpose of a product of matrices equals the product of their transposes in the reverse order.
If v1 and v2 are in ℝ4 and v2 is not a scalar multiple of v1, then {v1,v2} is linearly independent.
The statement is false. The vector v1 could be the zero vector.
If a set in ℝn is linearly dependent, then the set contains more vectors than there are entries in each vector.
The statement is false. There exists a set in ℝn that is linearly dependent and contains n vectors. One example is a set in ℝ2 consisting of two vectors where one of the vectors is a scalar multiple of the other.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
The statement is 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.
Two matrices are row equivalent if they have the same number of rows.
The statement is false. Two matrices are row equivalent if there exists a sequence of elementary row operations that transforms one matrix into the other.
An example of a linear combination of vectors v1 and v2 is the vector (1/2)v1.
The statement is true because (1/2)v1=(1/2)v1+0v2.
A homogeneous equation is always consistent.
The statement is true. A homogenous equation can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. Such a system Ax=0 always has at least one solution, namely, x=0. Thus a homogenous equation is always consistent.
A linear transformation is a special type of function.
The statement is true. A linear transformation is a function from ℝn to ℝm that assigns to each vector x in ℝn a vector T(x) in ℝm.
The equation Ax=b is homogeneous if the zero vector is a solution.
The statement is true. A system of linear equations is said to be homogeneous if it can be written in the form Ax=0, where A is an m×n matrix and 0 is the zero vector in ℝm. If the zero vector is a solution, then b=Ax=A0=0.
Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
The statement is true. Each elementary row operation replaces a system with an equivalent system.
Every matrix transformation is a linear transformation.
The statement is true. Every matrix transformation has the properties T(u+v)=T(u)+T(v) and T(cu)=cT(u) for all u and v, in the domain of T and all scalars c.
If A=abcd and ad=bc, then A is not invertible.
The statement is true. If ad=bc then ad−bc=0, and 1ad−bcd−b−ca is undefined.
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.
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. The second row of AB is the second row of A multiplied on the right by B.
The statement is true. Let rowi(A) denote the ith row of matrix A. Then rowi(AB)=rowi(A)B. Letting i=2 verifies this statement.
Reducing a matrix to echelon form is called the forward phase of the row reduction process.
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.
If A can be row reduced to the identity matrix, then A must be invertible.
The statement is true. Since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible.
If A is an invertible n×n matrix, then the equation Ax=b is consistent for each b in ℝn.
The statement is true. Since A is invertible, A−1b exists for all b in ℝn. Define x=A−1b. Then Ax=b.
If A is invertible, then the inverse of A−1 is A itself.
The statement is true. Since A−1 is the inverse of A, A−1A=I=AA−1. Since A−1A=I=AA−1, A is the inverse of A−1
If x and y are linearly independent, and if z is in Span{x, y}, then {x, y, z} is linearly dependent.
The statement is true. Since z is in Span{x, y}, z is a linear combination of x and y. Since z is a linear combination of x and y, the set {x, y, z} is linearly dependent.
Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Determine whether the statement below is true or false. Justify the answer.AB+AC=A(B+C)
The statement is true. The distributive law for matrices states that A(B+C)=AB+AC.
Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in Span a1, a2, a3.
The statement is true. The linear system corresponding to [a1 a2 a3 b] has a solution when b can be written as a linear combination of a1, a2, and a3. This is equivalent to saying that b is in Span a1, a2, a3
A transformation T is linear if and only if Tc1v1+c2v2=c1Tv1+c2Tv2 for all v1 and v2 in the domain of T and for all scalars c1 and c2.
The statement is true. This equation correctly summarizes the properties necessary for a transformation to be linear.
Two vectors are linearly dependent if and only if they lie on a line through the origin.
The statement is 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.
Any list of five real numbers is a vector in ℝ5.
The statement is true. ℝ5 denotes the collection of all lists of five real numbers.
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.
The equation Ax=b is consistent if the augmented matrix Ab has a pivot position in every row.
This statement is false. 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×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then A cannot have a pivot position in every row.
This statement is true. If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then the equation Ax=b has no solution for some b in ℝm.
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.
This statement is true. The equation Ax=b has the same solution set as the equation x1a1+x2a2+...+xnan=b
The first entry in the product Ax is a sum of products.
This statement is 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.
Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.
This statement is 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.
This statement is true. The matrix equation Ax=b is simply another notation for the vector equation x1a1+x2a2+ +xnan=b,where a1,...,an are the columns of A.
Every elementary row operation is reversible.
True. Replacement, Interchange, and Scaling are reversible.
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 x1a1+x2a2+x3a3=b.
True. The augmented matrix for x1a1+x2a2+x3a3=b is [a1 a2 a3 b].
Construct a 3×3 matrix A, with nonzero entries, and a vector b in ℝ3 such that b is not in the set spanned by the columns of A.
[ 1 1 1 ] [4] [ 2 2 2 ] [5] [ 3 3 3 ] [6]
