Deck 7: Vector Spaces

Question 1

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.

Answer 1

V is not a vector space, since it contains no zero vector.

Question 2

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.

Answer 2

V is a vector space.

Question 3

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.

Answer 3

S is a subspace.

Question 4

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

Question 5

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.

Question 6

In the vector space , let S be the set of all sequences such that there exists some such that for all (all eventually-zero sequences). Determine if S is a subspace of , and if not explain why.

Question 7

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.

Question 8

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.

Question 9

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

Question 10

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.

Question 11

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

Question 12

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

Question 13

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

Question 14

If then the polynomial space is a subspace of .

Question 15

If S is a subspace of the polynomial space , then for some m with .

Question 16

Determine if is in the subspace of given by .

Question 17

Determine if is in the subspace of given by .

Question 18

Determine if is in the subspace of given by .

Question 19

Determine if is in the subspace of given by .

Question 20

Determine if is in the subspace of given by .

Question 21

Determine if the set of vectors is linearly independent in .

Question 22

Determine if the set of vectors is linearly independent in .

Question 23

Determine if the set of vectors is linearly independent in .

Question 24

Determine if the set of vectors is linearly independent in P.

Question 25

Determine if the set of vectors spans P.

Question 26

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

Question 27

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

Question 28

If S is a linearly independent subset of a vector space V, but S does not span V, then there exists a vector v in V such that the set is also linearly independent.

Question 29

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.

Question 30

Every vector space V has a finite set S which both spans V and is linearly independent in V.

Question 31

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

Question 32

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

Question 33

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

Question 34

Find a basis for the subspace S of consisting of all polynomials p satisfying .

Question 35

Find a basis for the subspace S of consisting of all linear transformations such that
for some diagonal matrix D.

Question 36

Find a basis for the subspace S of P consisting of all even polynomials, that is, all polynomials p satisfying for all x.

Question 37

Determine the dimension of the subspace S of consisting of all symmetric matrices.

Question 38

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

Question 39

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

Question 40

Determine the dimension of the subspace S of consisting of all sequences such that the series is convergent and satisfies .

Question 41

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

Question 42

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

Question 43

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

Question 44

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

Question 45

If every infinite set S in a vector space V is linearly dependent, then .