Question 1

If S spans a subspace of a vector space V, then S is linearly independent in V.

Accepted Answer

A) True 
 B)False  False

Question 2

For every   , the set   of all polynomials of degree less than or equal to n is a subspace of   .

Accepted Answer

A) True 
 B)False  True

Question 3

If S spans a vector space V, but S is not linearly independent in V, then there exists a vector v in S such that the set difference   also spans V.

Accepted Answer

A) True 
 B)False  True

Question 4

Determine the dimension of the subspace S of   consisting of all matrices whose trace is 0.

Accepted Answer

The answer of Determine the dimension of the subspace S...

Question 5

Determine if the set of vectors   is linearly independent in   .

Accepted Answer

The set is

Question 6

The set V of all nonnegative real numbers, using the usual rules for vector addition and scalar multiplication in R, is a vector space.

Accepted Answer

A) True 
 B)False

Question 7

If   then the polynomial space   is a subspace of   .

Accepted Answer

A) True 
 B)False

Question 8

Determine the dimension of the subspace S of   consisting of all matrices A such that

Accepted Answer

The answer of Determine the dimension of the subspace S...

Question 9

Let V be the set of all vectors   , where x is in R. Using the usual rules for vector addition and scalar multiplication in R2, determine if V is a vector space, and if not explain why.

Accepted Answer

V is not a vector sp

Question 10

Determine if the given set is a basis for the vector space   .

Accepted Answer

The set is

Question 11

In the vector space   , let S be the set of all sequences   such that   . Determine if S is a subspace of   , and if not explain why.

Accepted Answer

The answer of In the vector space   ,...

Question 12

Determine if   is in the subspace of   given by   .

Accepted Answer

v is not i

Question 13

In the vector space   , let S be the set of matrices A such that   . Determine if S is a subspace of   , and if not explain why.

Accepted Answer

The answer of In the vector space   ,...

Question 14

If V is a vector space and   for all vectors v and w in V, then V consists of only the zero vector.

Accepted Answer

A) True 
 B)False

Question 15

Let V be the set of all functions f : R   R such that   . Using the usual rules for vector addition and scalar multiplication of functions, determine if V is a vector space, and if not explain why.

Accepted Answer

V is a vec

Question 16

Let V be a vector space with vector addition   and scalar multiplication   , and let W be a vector space with vector addition   and scalar multiplication   . Define     , with addition   and scalar multiplication   . Determine if   is a vector space, and if not explain why.

Accepted Answer

is a vecto

Question 17

If every finite set S in a vector space V fails to span V, then   .

Accepted Answer

A) True 
 B)False

Question 18

Determine if   is in the subspace of   given by   .

Accepted Answer

v is in th

Question 19

In the vector space   , let S be the set of all functions f such that   . Determine if S is a subspace, and if not explain why.

Accepted Answer

The answer of In the vector space   ,...

Question 20

If   and   are infinite-dimensional subspaces of a vector space V, then   .

Accepted Answer

A) True 
 B)False

If S spans a subspace of a vector space V, then S is linearly independent in V.

For every , the set of all polynomials of degree less than or equal to n is a subspace of .

If S spans a vector space V, but S is not linearly independent in V, then there exists a vector v in S such that the set difference also spans V.

Determine the dimension of the subspace S of consisting of all matrices whose trace is 0.

Determine if the set of vectors is linearly independent in .

The set V of all nonnegative real numbers, using the usual rules for vector addition and scalar multiplication in R, is a vector space.

If then the polynomial space is a subspace of .

Determine the dimension of the subspace S of consisting of all matrices A such that

Let V be the set of all vectors , where x is in R. Using the usual rules for vector addition and scalar multiplication in R², determine if V is a vector space, and if not explain why.

Determine if the given set is a basis for the vector space .

In the vector space , let S be the set of all sequences such that . Determine if S is a subspace of , and if not explain why.

Determine if is in the subspace of given by .

In the vector space , let S be the set of matrices A such that . Determine if S is a subspace of , and if not explain why.

If V is a vector space and for all vectors v and w in V, then V consists of only the zero vector.

Let V be the set of all functions f : R R such that . Using the usual rules for vector addition and scalar multiplication of functions, determine if V is a vector space, and if not explain why.

Let V be a vector space with vector addition and scalar multiplication , and let W be a vector space with vector addition and scalar multiplication . Define , with addition and scalar multiplication . Determine if is a vector space, and if not explain why.

If every finite set S in a vector space V fails to span V, then .

Determine if is in the subspace of given by .

In the vector space , let S be the set of all functions f such that . Determine if S is a subspace, and if not explain why.

If and are infinite-dimensional subspaces of a vector space V, then .

Systems of Linear Equations

Euclidean Space

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

Exam 7: Vector Spaces

If S spans a subspace of a vector space V, then S is linearly independent in V.

For every , the set of all polynomials of degree less than or equal to n is a subspace of .

If S spans a vector space V, but S is not linearly independent in V, then there exists a vector v in S such that the set difference also spans V.

Determine the dimension of the subspace S of consisting of all matrices whose trace is 0.

Determine if the set of vectors is linearly independent in .

The set V of all nonnegative real numbers, using the usual rules for vector addition and scalar multiplication in R, is a vector space.

If then the polynomial space is a subspace of .

Determine the dimension of the subspace S of consisting of all matrices A such that

Let V be the set of all vectors , where x is in R. Using the usual rules for vector addition and scalar multiplication in R2, determine if V is a vector space, and if not explain why.

Determine if the given set is a basis for the vector space .

In the vector space , let S be the set of all sequences such that . Determine if S is a subspace of , and if not explain why.

Determine if is in the subspace of given by .

In the vector space , let S be the set of matrices A such that . Determine if S is a subspace of , and if not explain why.

If V is a vector space and for all vectors v and w in V, then V consists of only the zero vector.

Let V be the set of all functions f : R R such that . Using the usual rules for vector addition and scalar multiplication of functions, determine if V is a vector space, and if not explain why.

Let V be a vector space with vector addition and scalar multiplication , and let W be a vector space with vector addition and scalar multiplication . Define , with addition and scalar multiplication . Determine if is a vector space, and if not explain why.

If every finite set S in a vector space V fails to span V, then .

Determine if is in the subspace of given by .

In the vector space , let S be the set of all functions f such that . Determine if S is a subspace, and if not explain why.

If and are infinite-dimensional subspaces of a vector space V, then .

Systems of Linear Equations

Euclidean Space

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Orthogonality

Linear Transformations

Inner Product Spaces

Additional Topics and Applications

Filters

Let V be the set of all vectors , where x is in R. Using the usual rules for vector addition and scalar multiplication in R², determine if V is a vector space, and if not explain why.