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Deck 7: Vector Spaces
ملء الشاشة (f)
سؤال
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.
, 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.
سؤال
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.
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. سؤال
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.
, let S be the set of all functions f such that . Determine if S is a subspace, and if not explain why. سؤال
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.
, let S be the set of all functions f such that . Determine if S is a subspace of V, and if not explain why. سؤال
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.
, let S be the set of all sequences such that . Determine if S is a subspace of , and if not explain why. سؤال
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.
, 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. سؤال
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.
, let S be the set of matrices A such that . Determine if S is a subspace of , and if not explain why. سؤال
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.
, let S be the set of matrices A such that . Determine if S is a subspace of , and if not explain why. سؤال
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.
, 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. سؤال
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.
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. سؤال
سؤال
If V is a vector space and 
for all vectors v and w in V, then V consists of only the zero vector.
for all vectors v and w in V, then V consists of only the zero vector. سؤال
For every 
, the set 
of all polynomials of degree less than or equal to n is a subspace of 
.
, the set of all polynomials of degree less than or equal to n is a subspace of . سؤال
If 
then the polynomial space 
is a subspace of 
.
then the polynomial space is a subspace of . سؤال
If S is a subspace of the polynomial space 
, then 
for some m with 
.
, then for some m with . سؤال
Determine if 
is in the subspace of 
given by 
.
is in the subspace of given by . سؤال
Determine if 
is in the subspace of 
given by 
.
is in the subspace of given by . سؤال
Determine if 
is in the subspace of 
given by 
.
is in the subspace of given by . سؤال
Determine if 
is in the subspace of 
given by 
.
is in the subspace of given by . سؤال
Determine if 
is in the subspace of 
given by 
.
is in the subspace of given by . سؤال
Determine if the set of vectors 
is linearly independent in 
.
is linearly independent in . سؤال
Determine if the set of vectors 
is linearly independent in 
.
is linearly independent in . سؤال
Determine if the set of vectors 
is linearly independent in 
.
is linearly independent in . سؤال
Determine if the set of vectors 
is linearly independent in P.
is linearly independent in P. سؤال
Determine if the set of vectors 
spans P.
spans P. سؤال
سؤال
سؤال
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.
is also linearly independent. سؤال
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.
also spans V. سؤال
سؤال
Determine if the set 
is a basis for the vector space 
.
is a basis for the vector space . سؤال
Determine if the given set is a basis for the vector space 
.
. سؤال
Determine if the given set is a basis for the vector space 
.
. سؤال
Find a basis for the subspace S of 
consisting of all polynomials p satisfying 
.
consisting of all polynomials p satisfying . سؤال
Find a basis for the subspace S of 
consisting of all linear transformations 
such that
for some 
diagonal matrix D.
consisting of all linear transformations such that for some diagonal matrix D. سؤال
Find a basis for the subspace S of P consisting of all even polynomials, that is, all polynomials p satisfying 
for all x.
for all x. سؤال
Determine the dimension of the subspace S of 
consisting of all symmetric matrices.
consisting of all symmetric matrices. سؤال
Determine the dimension of the subspace S of 
consisting of all matrices whose trace is 0.
consisting of all matrices whose trace is 0. سؤال
Determine the dimension of the subspace S of 
consisting of all matrices A such that
consisting of all matrices A such that سؤال
Determine the dimension of the subspace S of 
consisting of all sequences 
such that the series 
is convergent and satisfies 
.
consisting of all sequences such that the series is convergent and satisfies . سؤال
If 
and 
are subspaces of a vector space V and 
, then 
.
and are subspaces of a vector space V and , then . سؤال
If 
and 
are infinite-dimensional subspaces of a vector space V, then 
.
and are infinite-dimensional subspaces of a vector space V, then . سؤال
If 
and 
are infinite-dimensional subspaces of a vector space V, then 
is an infinite-dimensional subspace of V.
and are infinite-dimensional subspaces of a vector space V, then is an infinite-dimensional subspace of V. سؤال
If every finite set S in a vector space V fails to span V, then 
.
. سؤال
If every infinite set S in a vector space V is linearly dependent, then 
.
.
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ملء الشاشة (f)
Deck 7: Vector Spaces
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.
, 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.V is not a vector space, since it contains no zero vector.
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.
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.V is a vector space.
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.
, let S be the set of all functions f such that . Determine if S is a subspace, and if not explain why.S is a subspace.
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.
, let S be the set of all functions f such that . Determine if S is a subspace of V, and if not explain why. فتح الحزمة
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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.
, let S be the set of all sequences such that . Determine if S is a subspace of , and if not explain why. فتح الحزمة
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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.
, 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. فتح الحزمة
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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.
, let S be the set of matrices A such that . Determine if S is a subspace of , and if not explain why. فتح الحزمة
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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.
, let S be the set of matrices A such that . Determine if S is a subspace of , and if not explain why. فتح الحزمة
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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.
, 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. فتح الحزمة
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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.
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. فتح الحزمة
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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.
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12
If V is a vector space and for all vectors v and w in V, then V consists of only the zero vector.
for all vectors v and w in V, then V consists of only the zero vector. فتح الحزمة
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13
For every , the set of all polynomials of degree less than or equal to n is a subspace of .
, the set of all polynomials of degree less than or equal to n is a subspace of . فتح الحزمة
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14
If then the polynomial space is a subspace of .
then the polynomial space is a subspace of . فتح الحزمة
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15
If S is a subspace of the polynomial space , then for some m with .
, then for some m with . فتح الحزمة
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16
Determine if is in the subspace of given by .
is in the subspace of given by . فتح الحزمة
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17
Determine if is in the subspace of given by .
is in the subspace of given by . فتح الحزمة
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18
Determine if is in the subspace of given by .
is in the subspace of given by . فتح الحزمة
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19
Determine if is in the subspace of given by .
is in the subspace of given by . فتح الحزمة
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20
Determine if is in the subspace of given by .
is in the subspace of given by . فتح الحزمة
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21
Determine if the set of vectors is linearly independent in .
is linearly independent in . فتح الحزمة
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22
Determine if the set of vectors is linearly independent in .
is linearly independent in . فتح الحزمة
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23
Determine if the set of vectors is linearly independent in .
is linearly independent in . فتح الحزمة
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24
Determine if the set of vectors is linearly independent in P.
is linearly independent in P. فتح الحزمة
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25
Determine if the set of vectors spans P.
spans P. فتح الحزمة
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26
If S spans a subspace of a vector space V, then S is linearly independent in V.
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27
If S is a linearly independent subset of a vector space V, then S spans V.
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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.
is also linearly independent. فتح الحزمة
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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.
also spans V. فتح الحزمة
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30
Every vector space V has a finite set S which both spans V and is linearly independent in V.
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31
Determine if the set is a basis for the vector space .
is a basis for the vector space . فتح الحزمة
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32
Determine if the given set is a basis for the vector space .
. فتح الحزمة
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33
Determine if the given set is a basis for the vector space .
. فتح الحزمة
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34
Find a basis for the subspace S of consisting of all polynomials p satisfying .
consisting of all polynomials p satisfying . فتح الحزمة
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35
Find a basis for the subspace S of consisting of all linear transformations such that
for some diagonal matrix D.
consisting of all linear transformations such that for some diagonal matrix D. فتح الحزمة
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36
Find a basis for the subspace S of P consisting of all even polynomials, that is, all polynomials p satisfying for all x.
for all x. فتح الحزمة
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37
Determine the dimension of the subspace S of consisting of all symmetric matrices.
consisting of all symmetric matrices. فتح الحزمة
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38
Determine the dimension of the subspace S of consisting of all matrices whose trace is 0.
consisting of all matrices whose trace is 0. فتح الحزمة
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39
Determine the dimension of the subspace S of consisting of all matrices A such that
consisting of all matrices A such that فتح الحزمة
افتح القفل للوصول البطاقات البالغ عددها 45 في هذه المجموعة.
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40
Determine the dimension of the subspace S of consisting of all sequences such that the series is convergent and satisfies .
consisting of all sequences such that the series is convergent and satisfies . فتح الحزمة
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41
If and are subspaces of a vector space V and , then .
and are subspaces of a vector space V and , then . فتح الحزمة
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42
If and are infinite-dimensional subspaces of a vector space V, then .
and are infinite-dimensional subspaces of a vector space V, then . فتح الحزمة
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43
If and are infinite-dimensional subspaces of a vector space V, then is an infinite-dimensional subspace of V.
and are infinite-dimensional subspaces of a vector space V, then is an infinite-dimensional subspace of V. فتح الحزمة
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44
If every finite set S in a vector space V fails to span V, then .
. فتح الحزمة
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45
If every infinite set S in a vector space V is linearly dependent, then .
. فتح الحزمة
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