Linear Algbera 2 Flashcards
State the row operation performed below and describe how it affects the determinant.
[a b] [4 a 4b]
[c d] [c d ]
A. The row operation scales row 1 by one-fourth
B. The row operation subtracts 4 from row 1
C. The row operation scales row 1 by 4
D. The row operation adds 4 to row 1
How does this affect the determinant
The determinant is 0.
B. The determinant is multiplied by 4.
C. The determinant is divided by 4.
D. The determinant is unchanged.
C, B
Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
[3 7] [3 7]
[8 1] , [8+3k 1 +7k]
A. Replace row 2 with k times row 1.
B. Replace row 2 with k times row 2.
C. Replace row 2 with k times row 1 plus row 2.
D. Replace row 2 with row 1 plus k times row 2.
How does the row operation affect the determinant?
A. The determinant is increased by 42k.
B. The determinant is decreased by 21k.
C. The determinant is increased by 21k.
D.The determinant does not change.
C,D
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
[3 6 4] [3 6 4]
[a b c] [5 9 7]
[5 9 7] [a b c]
What is the elementary row operation?
A. Row 3 is replaced with the sum of rows 1 and 3.
B. Rows 1 and 3 are interchanged.
C. Row 2 is replaced with the sum of rows 2 and 3.
D. Rows 1 and 2 are interchanged.
E. Row 2 is replaced with the sum of rows 1 and 2.
F. Row 3 is replaced with the sum of rows 2 and 3.
G. Rows 2 and 3 are interchanged.
Part 2
How does the row operation affect the determinant?
A. It multiplies the determinant by 2.
B. It increases the determinant by 1.
C. It changes the sign of the determinant.
D. It does not change the determinant.
G, C
Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
[-5 -7 -3] [-5 7 -3]
[4 -2 4] [4 -2 4]
[1 1 1] [k k k ]
What is the elementary row operation?
A. Replace row 3 with k plus row 3.
B. Replace row 3 with k times row 3.
C. Replace row 3 with row 3 divided by k.
D. Replace row 3 with row 3 minus k.
Part 2
How does the row operation affect the determinant?
A. The determinant is decreased by 3k.
B. The determinant is multiplied by k.
C. The determinant is increased by 3k.
D. The determinant does not change.
B, B
Let A be an n*n matrix. Determine whether the statement below is true or false. Justify the answer.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
Question content area bottom
Part 1
Choose the correct answer below.
A. The statement is true. Cofactor expansion along the row (or column) with the most zeros of a triangular matrix produces a determinant equal to the sum of the entries along the main diagonal.
B. The statement is false. The determinant of a matrix is the arithmetic mean of the entries along the main diagonal.
C. The statement is false. The determinant of a triangular matrix is the product of the entries along the main diagonal.
C
State which property of determinants is illustrated in this equation.
[-9 8 -4] [27 -6 0]
[27 -6 0] [-9 8 -4]
[2 3 -5] [2 3 -5]
A. If two rows of A are interchanged to produce B, then detB= -det A.
B. If one row of A is multiplied by k to produce B, then det B = k *det A.
C. If a multiple of one row of A is added to another row to produce matrix B, then detB equals det A.
D. If A and B are square matrices, then detAB = (det A)(det B)
A
The equation below illustrates a property of determinants. State the property.
[3 -6 9] [1 -2 3]
[3 5 -5] 3 *[3 5 -5]
[1 3 3] [1 3 3]
A. Multiplying a row by 3 divides the determinant by 3.
B. Dividing a row by 3 multiplies the determinant by 3.
C. Multiplying a row by 3 multiplies the determinant by 3.
D. Factoring 3 out of the entire matrix divides the determinant by 3.
C
State which property of determinants is illustrated in this equation.
[ 8 1 -4] [8 1 -4]
[-24 -2 5] [0 1 -7]
[0 8 -3] [0 8 -3]
A If two rows of A are interchanged to produce B, then detB = - det A.
B. If A and B are square matrices, then det AB = (det A)(det B)
C. If a multiple of one row of A is added to another row to produce matrix B, then det B equals detA.
D. If one row of A is multiplied by k to produce B, then det B = k*detA
C
Let A be an n*n matrix. Determine whether the statement below is true or false. Justify the answer.
If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
A. The statement is false. If A= [2 6]
[1 3]
, then det A=0 and the rows and columns are all distinct and not full of zeros.
B. The statement is true. If det A is zero, then the columns of A are linearly independent. If one column is zero, or two columns are the same, then the columns are linearly dependent.
C. The statement is false. The determinant depends on the columns of A. It is possible for two rows to be the same and for the determinant to be nonzero.
D.
The statement is true. If A = [2 3]
[2 3]
and B =[1 2]
[0 0]
then detA = 0 and detB =0
A
Let A and B be n*n matrices. Determine whether the statement below is true or false. Justify the answer.
det(A+B)=det A+det B
Choose the correct answer below.
A. The statement is false. If A= [ 1 0]
[0 1]
and B = [-1 0]
[0 -1] then det(A+B) = 0 then detA + detB =2
B. The statement is true. Determinants are linear transformations.
C. The statement is false. det(A+ B)=(det A)(det B)
D. The statement is true. If A=[2 0]
[1 0]
and B = [3 0]
[5 0] then det(A+B)= 0 and detA+detB=0
A
Determine if the given set is a subspace of set of prime numbers P 6. Justify your answer.
The set of all polynomials of the form p(t)=at^6, where a is in set of real numbers R.
Choose the correct answer below.
A.
The set is a subspace of set of prime numbers P 6. The set contains the zero vector of set of prime numbers P 6, the set is closed under vector addition, and the set is closed under multiplication on the left by m*6 matrices where m is any positive integer.
B. The set is a subspace of set of prime numbers P 6. The set contains the zero vector of set of prime numbers P 6, the set is closed under vector addition, and the set is closed under multiplication by scalars.
C. The set is not a subspace of set of prime numbers P 6. The set is not closed under multiplication by scalars when the scalar is not an integer.
D.The set is not a subspace of set of prime numbers P 6. The set does not contain the zero vector of set of prime numbers P 6
B
Determine if the given set is a subspace of set of prime numbers P_n. Justify your answer.
The set of all polynomials in set of prime numbers P_n such that p(0)=0
Choose the correct answer below.
A.
The set is not a subspace of set of prime numbers P_n because the set is not closed under vector addition.
B. The set is a subspace of set of prime numbers P_n because set of prime numbers P_n is a vector space spanned by the given set.
C. The set is not a subspace of set of prime numbers P_n because the set is not closed under multiplication by scalars.
D. The set is a subspace of set of prime numbers P_n because the set contains the zero vector of set of prime numbers P_n, the set is closed under vector addition, and the set is closed under multiplication by scalars.
E. The set is not a subspace of set of prime numbers P_n because the set does not contain the zero vector of set of prime numbers P_n.
D
Determine whether the statement is True or False. Justify your answer.
If u is a vector in a vector space V, then (minus1)u is the same as the negative of u.
Choose the correct answer below.
A. The statement is true. For each u in V, there is a vector -u in V such that u+(-u)=0
B. The statement is true. For each u in V, there is a vector -u in V such that -u = -1u.
C. The statement is false. For each u in V, there is a vector -u in V such that u+-u = 0.
D. The statement is false. For each u in V, there is a vector -u in V such that -u=u.
B
A denotes an m*n matrix. Determine whether the statement is true or false. Justify your answer.
The null space of A, Nul A, is the kernel of the mapping x->Ax.
A. The statement is false. The kernel of a linear transformation T, from a vector space V to a vector space W, is the set of all u in V such that T(u)=0. Thus, the kernel of a matrix transformation T(x)=Ax is the column space of A, not the null space of A.
B. The statement is false. The kernel of a linear transformation T, from a vector space V to a vector space W, is the set of all vectors in W of the form T(x) for some x in V. Thus, the kernel of a matrix transformation T(x)= Ax is the column space of A, not the null space of A.
C. The statement is true. The kernel of a linear transformation T, from a vector space V to a vector space W, is the set of all vectors in W of the form T(x) for some x in V. Thus, the kernel of a matrix transformation T(x)=Ax is the null space of A.
D. The statement is true. The kernel of a linear transformation T, from a vector space V to a vector space W, is the set of all u in V such that T(u)=0. Thus, the kernel of a matrix transformation T(x)=Ax is the null space of A.
D
A denotes an m*n matrix. Determine whether the statement is true or false. Justify your answer.
The row space of A^T is the same as the column space of A.
Choose the correct answer below.
A. The statement is true because the rows of A^ T are the columns of (A^T)^T
B. The statement is false because the number of free variables in the equation A^Tx=0 is the same as the number of pivot columns of (A^T)^T=A
C. The statement is false because the rows of A^T are also the rows of (A^T)^T=A
D. The statement is true because the number of pivot columns of A^ T is the same as the number of pivot columns of (A^T)^T=A
A
Let A be an m*n matrix. Determine whether the statement below is true or false. Justify the answer.
The number of variables in the equation Ax=0 equals the nullity of A.
A. This statement is false. The number of free variables is equal to the nullity of A.
B. This statement is true. The number of variables in the equation Ax=0 equals the number of columns in A, and the number of columns in A equals the nullity of A.
C. This statement is true. The nullity of A equals the largest number of 0s in any solution to the equation Ax=b, and the equation Ax=0 always has the trivial solution, so the number of variables equals the nullity of A.
D. This statement is false. The number of pivot columns is equal to the nullity of A.
A
Let A be an mtimesn matrix. Determine whether the statement below is true or false. Justify the answer.
The nullity of A is the number of columns of A that are not pivot columns.
Choose the correct answer below.
A. The statement is false. The dimension of the column space of A is the number of columns of A that are not pivot columns.
B. The statement is true. The nullity of A equals the number of columns of A minus the number of free variables in the equation Ax=0.
C. The statement is true. The nullity of A equals the number of free variables in the equation Ax=0.
D. The statement is false. The nullity of A equals the number of pivot columns.
C
Suppose a 7*11 matrix A has three pivot columns. Is Col A=R^3. Is Nul Upper A= R^8? Explain your answers.
Is Col A=R^3
A. Yes. Since A has three pivot columns, dim Col A = 3. Thus, Col A is a three-dimensional subspace of R^3, so Col A is equal to R^3
B. No, Col A is not R^3. Since A has three pivot columns, dim Col A=8. Thus, Col A is equal to R^8
C. No, the column space of A is not R^3. Since A has three pivot columns, dim Col A = 4. Thus, Col A is equal to R^4.
D. No. Since A has three pivot columns, dim Col A=3. But Col A is a three-dimensional subspace of R^7, so Col A is not equal to R^3
Part 2
Is Nul A =R^8
A. No, Nul A is not equal to set of real numbers R^8. It is true that dim Nul A=8, but Nul A is a subspace of R^11.
B. No, Nul A is equal to R^8. Since A has three pivot columns, dim Nul A=4. Thus, Nul A is equal to R^4.
C. No, Nul A is not equal to R^8. Since A has three pivot columns, dim Nul A=3. Thus, Nul A is equal to R^3
D. Yes, Nul A is equal to R^8.. Since A has three pivot columns, dim Nul A=8. Thus, Nul A is equal to R^8
D,A
If the nullity of a 4*8 matrix A is 4, what are the dimensions of the column and row spaces of A?
dim Col A=
dim Row A=
4
4
If A is a 75 matrix, what is the largest possible rank of A? If A is a 57 matrix, what is the largest possible rank of A? Explain your answers.
A. The rank of A is equal to the number of pivot positions in A. Since there are only __ columns in a 75 matrix, and there are only __ rows in a 57 matrix, there can be at most __ pivot positions for either matrix. Therefore, the largest possible rank of either matrix is __
B.
The rank of A is equal to the number of columns of A. Since there are __ columns in a ___matrix, the largest possible rank of a __ matrix is __. Since there are __ columns in a ___ matrix, the largest possible rank of a ____ matrix is __
C. The rank of A is equal to the number of non-pivot columns in A. Since there are more rows than columns in a 75 matrix, the rank of a 75 matrix must be equal to __. Since there are 5 rows in a 57 matrix, there are a maximum of 5 pivot positions in A. Thus, there are 2 non-pivot columns. Therefore, the largest possible rank of a 57 matrix is __
A
5
5
If A is a 96 matrix, what is the largest possible dimension of the row space of A? If A is a 69 matrix, what is the largest possible dimension of the row space of A? Explain.
Select the correct choice below and fill in the answer box(es) to complete your choice.
A. The dimension of the row space of A is equal to the number of non-pivot columns in A. Since there are more rows than columns in a 96 matrix, the dimension of the row space of a 96 matrix must equal
__ . Since there are 6 rows in a 69 matrix, there are a maximum of 6 pivot positions in A and 3 non-pivot columns. Therefore, the largest possible dimension of the row space of a 69 matrix is __
B. The dimension of the row space of A is equal to the number of pivot positions in A. Since there are only 6 columns in a 96 matrix, and there are only 6 rows in a 69 matrix, there can be at most __
pivot positions for either matrix. Therefore, the largest possible dimension of the row space of either matrix is __
C.The dimension of the row space of A is equal to the number of rows of A, which is equal to the number of pivot positions in A. Since there are 9 rows in a 96 matrix, the largest possible dimension of the row space of a 96 matrix is __. Since there are 6 rows in a 69 matrix, the largest possible dimension of the row space of a 69 matrix is _
B
6
6
Let V be a nonzero finite-dimensional vector space, and the vectors listed belong to V. Determine whether the statement below is true or false. Justify the answer.
If there exists a set { v 1, … ,v_ p} that spans V, then dim V<=p
Part 1
Choose the correct answer below.
A.
This statement is false. The set { v 1, … , v_p} can be considered a subspace of a finite-dimensional vector space, which can be expanded to find the basis for V. This basis will have at least p elements, so dim V>=p
B. This statement is false. Apply the Spanning Set Theorem to the set { v 1, … , v_p} and produce a basis for V. This basis will have more than p elements in it, so dim V > p.
C. This statement is true. Apply the Spanning Set Theorem to the set { v 1, … , v_ p} and produce a basis for V. This basis will not have more than p elements in it, so dim V<=p
D. This statement is true. The set { v 1, … , v_ p} can be considered a subspace of a finite-dimensional vector space, which can be expanded to find the basis for V. This basis will have less than p elements, so dim V < p. If dim V < p is true then dim V <=p must also be true
C
Let V be a nonzero finite-dimensional vector space, and the vectors listed belong to V. Determine whether the statement below is true or false. Justify the answer.
If there exists a linearly independent set { v 1, … , v_ p} in V, then dim V>=p
Choose the correct answer below.
A.
This statement is true. Apply the Spanning Set Theorem to the set { v 1, … , v_ p} and produce a basis for V. This basis will have at least p elements in it, so dim V>=p.
B.
This statement is true. The span of the set { v 1, … , v_ p} will be a subspace of V which can be expanded to find a basis for V. This basis will contain at least p elements, so dim V>=p
C.
This statement is false. The span of the set {v 1, … , v_ p} will be a subspace of V which can be trimmed to find a basis for V. This basis will contain at most p elements, so dim V<=p
D.
This statement is false. Because the set { v 1, … , v p} is linearly independent, the dimension of V cannot be more than the number of elements in { v 1, … , v_p}, so dim V<=p
