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Linear Algebra HW Q for Test 1 Material
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Gravity
Terms in this set (72)
A, B, C, D, E, F
Let u1=[4,4], and u2=[−12,−7]. Select all of the vectors that are in the span of {u1,u2}. (Choose every statement that is correct.)
A. The vector [0,0] is in the span.
B. The vector −3[4,4] is in the span.
C. The vector 6[−12,−7]−3[4,4] is in the span.
D. The vector [−12,−7] is in the span.
E. All vectors in R2 are in the span.
F. The vector [4,4] is in the span.
G. We cannot tell which vectors are in the span.
F
Determine if the subset of R^2 consisting of vectors of the form [a,b], where a+b=1 is a subspace. T/F This set is closed under scalar multiplications
F
Determine if the subset of R^2 consisting of vectors of the form [a,b], where a+b=1 is a subspace. T/F This set is a subspace.
F
Determine if the subset of R^2 consisting of vectors of the form [a,b], where a+b=1 is a subspace. T/F This set is closed under vector addition
F
Determine if the subset of R^2 consisting of vectors of the form [a,b], where a+b=1 is a subspace. T/F The set contains the zero vector
T
Determine if the subset of R2 consisting of vectors of the form [a,b], where a and b are integers, is a subspace. T/F This set is closed under vector addition
F
Determine if the subset of R2 consisting of vectors of the form [a,b], where a and b are integers, is a subspace. T/F This set is a subspace
T
Determine if the subset of R2 consisting of vectors of the form [ab], where a and b are integers, is a subspace. T/F The set contains the zero vector.
F
Determine if the subset of R2 consisting of vectors of the form [ab], where a and b are integers, is a subspace. T/F This set is closed under scalar multiplications
F
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where abc=0 is a subspace.T/F This set is a subspace
T
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where abc=0 is a subspace.T/F The set contains the zero vector
T
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where abc=0 is a subspace.T/F This set is closed under scalar multiplications
F
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where abc=0 is a subspace.T/F This set is closed under vector addition
T
Determine if the subset of R3 consisting of vectors of the form <a,b,c> where a≥0, b≥0, and c≥0 is a subspace. T/F The set contains the zero vector
F
Determine if the subset of R3 consisting of vectors of the form <a,b,c> where a≥0, b≥0, and c≥0 is a subspace. T/F This set is a subspace
T
Determine if the subset of R3 consisting of vectors of the form <a,b,c> where a≥0, b≥0, and c≥0 is a subspace. T/F This set is closed under vector addition
F
Determine if the subset of R3 consisting of vectors of the form <a,b,c> where a≥0, b≥0, and c≥0 is a subspace. T/F This set is closed under scalar multiplications
T
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where at most one of a, b, and c is nonzero, is a subspace. T/F This set is closed under scalar multiplications
T
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where at most one of a, b, and c is nonzero, is a subspace. T/F The set contains the zero vector
Solution: T
F
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where at most one of a, b, and c is nonzero, is a subspace. T/F This set is closed under vector addition
F
Determine if the subset of R3 consisting of vectors of the form <a,b,c>, where at most one of a, b, and c is nonzero, is a subspace. T/F This set is a subspace
T
Determine if the subset of R2 consisting of vectors of the form <v1, ... , vn>, where v1−v2+v3−v4+v5−⋯−vn=0 is a subspace.T/F This set is closed under vector addition
T
Determine if the subset of R2 consisting of vectors of the form <v1, ... , vn>, where v1−v2+v3−v4+v5−⋯−vn=0 is a subspace.T/F This set is a subspace
T
Determine if the subset of R2 consisting of vectors of the form <v1, ... , vn>, where v1−v2+v3−v4+v5−⋯−vn=0 is a subspace.T/F The set contains the zero vector
T
Determine if the subset of R2 consisting of vectors of the form <v1, ... , vn>, where v1−v2+v3−v4+v5−⋯−vn=0 is a subspace.T/F This set is closed under scalar multiplications
F
If A is an n×n matrix and b≠0 in Rn, then consider the set of solutions to Ax=b. T/F This set is closed under vector addition
F
If A is an n×n matrix and b≠0 in Rn, then consider the set of solutions to Ax=b. T/F This set is a subspace
F
If A is an n×n matrix and b≠0 in Rn, then consider the set of solutions to Ax=b. T/F The set contains the zero vector
F
If A is an n×n matrix and b≠0 in Rn, then consider the set of solutions to Ax=b. T/F This set is closed under scalar multiplications
ACF
Which of the following subsets of R3×3 are subspaces of R3×3?
A. The symmetric 3×3 matrices
B. The 3×3 matrices whose entries are all integers
C. The 3×3 matrices with all zeros in the second row
D. The 3×3 matrices with determinant 0
E. The 3×3 matrices in reduced row-echelon form
F. The diagonal 3×3 matrices
BCFG
Determine whether the given set S is a subspace of the vector space V.
A. V=P2, and S is the subset of P2 consisting of all polynomials of the form p(x)=x2+c.
B. V=Rn, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
C. V=C2(I), and S is the subset of V consisting of those functions satisfying the differential equation y′′−4y′+3y=0.
D. V=C3(I), and S is the subset of V consisting of those functions satisfying the differential equation y′′′−y′=1.
E. V=R4, and S is the set of vectors of the form (0,x2,5,x4).
F. V=Mn(R), and S is the subset of all symmetric matrices
G. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=f(b).
Y
Determine if each of the following sets is a subspace of Pn, for an appropriate value of n. Y/N Let W1 be the set of all polynomials of the form p(t)=at2, where a is in R.
N
Determine if each of the following sets is a subspace of Pn, for an appropriate value of n. Y/N Let W2 be the set of all polynomials of the form p(t)=t2+a, where a is in R.
Y
Determine if each of the following sets is a subspace of Pn, for an appropriate value of n. Y/N Let W3 be the set of all polynomials of the form p(t)=at2+at, where a is in R.
E
Let u=<−1, −3, 3> v=<0, 0, 0>, and w=<−5, −11, 27>. We want to determine by inspection (with minimal computation) if {u,v,w} is linearly dependent or independent. Choose the best answer.
A. The set is linearly dependent because the number of vectors in the set is greater than the dimension of the vector space.
B. The set is linearly independent because we only have two vectors and they are not scalar multiples of each other.
C. The set is linearly dependent because one of the vectors is a scalar multiple of another vector.
D. The set is linearly dependent because two of the vectors are the same.
E. The set is linearly dependent because one of the vectors is the zero vector.
F. We cannot easily tell if the set is linearly dependent or linearly independent.
D
Let u= <4,−1, 1, 2>, v=<4, −1, 1, 2> and w=<−8, −2, 2, −2>. We want to determine by inspection (with minimal computation) if {u,v,w} is linearly dependent or independent. Choose the best answer.
A. The set is linearly dependent because one of the vectors is the zero vector.
B. The set is linearly independent because we only have two vectors and they are not scalar multiples of each other.
C. The set is linearly dependent because the number of vectors in the set is greater than the dimension of the vector space.
D. The set is linearly dependent because one of the vectors is a scalar multiple of another vector.
E. We cannot easily tell if the set is linearly dependent or linearly independent.
C
Let S be a set of m vectors in Rn with m>n. Select the best statement.
A. The set S is linearly independent, as long as it does not include the zero vector.
B. The set S could be either linearly dependent or linearly independent, depending on the case.
C. The set S is linearly dependent.
D. The set S is linearly independent, as long as no vector in S is a scalar multiple of another vector in the set.
E. The set S is linearly independent.
D
Let u=<4,−1,1,2>, v=<4,−1,1,2>, and w=<−8,−2,2,−2>. We want to determine by inspection (with minimal computation) if {u,v,w} is linearly dependent or independent. Choose the best answer.
A. The set is linearly dependent because one of the vectors is the zero vector.
B. The set is linearly independent because we only have two vectors and they are not scalar multiples of each other.
C. The set is linearly dependent because the number of vectors in the set is greater than the dimension of the vector space.
D. The set is linearly dependent because one of the vectors is a scalar multiple of another vector.
E. We cannot easily tell if the set is linearly dependent or linearly independent.
C
Let S be a set of m vectors in Rn with m>n. Select the best statement.
A. The set S is linearly independent, as long as it does not include the zero vector.
B. The set S could be either linearly dependent or linearly independent, depending on the case.
C. The set S is linearly dependent.
D. The set S is linearly independent, as long as no vector in S is a scalar multiple of another vector in the set.
E. The set S is linearly independent.
F. none of the above
B
Let A be a matrix with more rows than columns. Select the best statement.
A. The columns of A must be linearly independent.
B. The columns of A could be either linearly dependent or linearly independent depending on the case.
C. The columns of A are linearly independent, as long as they does not include the zero vector.
D. The columns of A are linearly independent, as long as no column is a scalar multiple of another column in A
E. The columns of A must be linearly dependent.
F. none of the above
A
Let A be a matrix with more columns than rows. Select the best statement.
A. The columns of A must be linearly dependent.
B. The columns of A are linearly independent, as long as no column is a scalar multiple of another column in A
C. The columns of A could be either linearly dependent or linearly independent depending on the case.
D. The columns of A are linearly independent, as long as they does not include the zero vector.
E. none of the above
Y
Determine whether each set {p1,p2} is a linearly independent set in P^3. Type "Y" or "N" for each answer.
The polynomials p1(t)=1+t2 and p2(t)=1−t2
Y
Determine whether each set {p1,p2} is a linearly independent set in P^3. Y/N The polynomials p1(t)=2t+t2 and p2(t)=1+t.
N
Determine whether each set {p1,p2} is a linearly independent set in P^3. Y/N The polynomials p1(t)=2t−4t2 and p2(t)=6t2−3t.
C
Let W be the set: <-2, 3, 0>, <6, -1,5>. Determine if W is a basis for R^3 and check the correct answer(s) below.
A. W is not a basis because it is linearly dependent.
B. W is a basis.
C. W is not a basis because it does not span R^3.
F
T/F The set {0} forms a basis for the zero subspace.
T
T/F If {u1,u2,u3} is a basis for R3, then span{u1,u2} is a plane.
T
T/F If the set of vectors U is linearly independent in a subspace S but is not a basis for S, then vectors can be added to U to create a basis for S
F
T/F If S1 and S2 are subspaces of Rn of the same dimension, then S1=S2.
T
T/F If the set of vectors U spans a subspace S but is not a basis for S, then vectors can be removed from U to create a basis for S
F
T/F Three nonzero vectors that lie in a plane in R3 might form a basis for R3.
F
T/F If the set of vectors U spans a subspace S, then vectors can be added to U to create a basis for S
2
Find the dim for the vector space R2
18
Find the dim for the vector space R6×3
10
Find the dim for the vector space of all upper triangular 4×4 matrices
24
Find the dim for the vector space of 5×5 matrices with trace 0.
7
Find the dim for the vector space P7[x] of polynomials with degree less than 7 .
5
Find the dim for the vector space of all diagonal 5×5 matrices .
T
T/F If {u1,u2,u3} is a basis for R3, then span{u1,u2} is a plane.
F
T/F Let m<n. Then U= {u1,u2,...,um} in Rn can form a basis for Rn if the correct n−m vectors are added to U.
F
T/F Let m>n. Then U= {u1,u2,...,um} in Rn can form a basis for Rn if the correct m−n vectors are removed from U.
F
T/F If S1 and S2 are subspaces of R^n with the same dimension, then S1=S2.
T
T/F If A and B are equivalent matrices, then row(A) = row(B).
F
T/F If A and B are equivalent matrices, then col(A) = col(B).
T
T/F If A is a matrix, then the dimension of the row space of A is equal to the dimension of the column space of A
3
Suppose that A is a 3×9 matrix that has an echelon form with no zero rows The dimension of the row space of A is?
3
Suppose that A is a 3×9 matrix that has an echelon form with no zero rows. The dimension of the column space of A is ?
6
Suppose that A is a 3×9 matrix that has an echelon form with no zero rows. The dimension of the solution space to Ax =0 is ?
3
If A is a 3×5 matrix, then the number of leading 1's in the reduced row echelon form of A is at most _________.
5
If A is a 3×5 matrix, then the number of parameters (free variables) in the general solution of Ax =0 is at most _________
3
If A is a 5×3 matrix, then the number of leading 1's in the reduced row echelon form of A is at most _________.
3
If A is a 5×3 matrix, then the number of parameters (free variables) in the general solution of Ax=0 is at most _________.
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