Abstract algebra II (MAT-602)-Semester-VI, Class: -T. Y. B.Sc.

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Dear students, this quiz is designed to test your knowledge of Abstract algebra II. You can attempt this quiz any number of times. For any queries contact me on my email ashokgodse2012@gmail.com or WhatsApp number 9767046600

Regards
A. D. Godase,
Head & Assistant Professor in Mathematics,
Vinayakrao Patil Mahavidyalaya Vaijapur,
Aurangabad-423701, India, (MH)

Thank you

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1. Let L[S] be the subspace spanned by the set. Then L[L(S)] *

1 point

L[S]

L'[S]

2. R is not a vector space over C *

1 point

True

False

3. Let T: V → W be a linear transformation then T is an epimorphism if

1 point

T is 1-1

T is 1-1 and onto

T is not 1-1

T is onto

Clear selection

4. A linear transformation T: V → W is called non-singular if T is *

1 point

1-1

1-1 and onto

onto

5. C is the vector space of dimension 2 over R. *

1 point

true

false

6. Any vector space of dimension n over a field F is isomorphic to *

1 point

Zn(F)

Vn(F)

Zn(R)

Vn(R)

7. Let T:V→ W be a linear transformation then T is a homomorphism if and only if *

1 point

ker T={0}

ker T={1}

ker T={e}

ker T={Φ}

8. What is the dimension of the vector space C? *

1 point

9.10. Let V be a five-dimensional vector space, and let S be a subset of V consisting of five vectors. Then S *

1 point

Must be linearly dependent, and must span V.

Must be linearly dependent, but may or may not span V.

Cannot span V, but can be linearly independent or dependent.

Must be a basis of V.

Must be linearly independent, but cannot span V.

Can span V, but only if it is linearly independent, and vice versa.

Must be linearly independent, but may or may not span V.

10. Let V be a three-dimensional vector space, and let V be a subset of V consisting of five vectors. Then S *

1 point

Must be linearly independent, but cannot span V.

May or may not be linearly independent, and may or may not span V.

Cannot span V, but can be linearly independent or dependent.

Can span V, but only if it is linearly independent, and vice versa.

Must be linearly independent, but may or may not span V.

Must be linearly dependent, and must span V.

Must be linearly dependent, but may or may not span V.

11. Let V be a five-dimensional vector space, and let V be a subset of V consisting of three vectors. Then S *

1 point

Cannot span V, but can be linearly independent or dependent.

Can span V, but only if it is linearly independent, and vice versa.

May or may not be linearly independent, and may or may not span V.

Must be linearly independent, but cannot span V.

Must be linearly dependent, but may or may not span V.

Must be linearly independent, but may or may not span V.

Must be linearly dependent, and must span V.

12. Let V be a five-dimensional vector space, and let S be a subset of V which is a basis for V. Then S *

1 point

Can have any number of elements (except zero).

Must have exactly five elements.

Must span V.

Must consist of at least five elements.

Must be linearly dependent.

Must be linearly independent.

Must have at most five elements.

13. Let V be a five-dimensional vector space, and let S be a subset of V which is linearly dependent. Then S *

1 point

Must have at most five elements.

Must span V.

Must have infinitely many elements.

Must be a basis for V.

Must have exactly five elements.

Must consist of at least five elements.

Can have any number of elements (except zero).

14. Let S be a five-dimensional vector space, and let S be a subset of V which is linearly independent. Then S *

1 point

Must span V.

Must have infinitely many elements.

Can have any number of elements (except zero).

Must be a basis for V.

Must have at most five elements.

Must consist of at least five elements.

Must have exactly five elements.

15. Let V be a five-dimensional vector space, and let S be a subset of V which spans V. Then S *

1 point

Must have at most five elements.

Must be a basis for V.

Must consist of at least five elements.

Must have infinitely many elements.

Must be linearly dependent.

Must have exactly five elements.

Must be linearly independent.

16. Let V be a vector space, and let S be a subset of V. What does it mean when we say that spans ? *

1 point

The elements of S are all distinct from each other.

The rank of S is the same as the dimension of V.

S has at least as many elements as the dimension of V.

Every vector in V can be expressed as a linear combination of vectors in S.

S is a basis for V.

Every vector in V has exactly one representation as a linear combination of vectors in S.

17. Let V be a vector space, and let S be a subset of V. What does it mean when we say that S is linearly dependent? *

1 point

S is closed under both addition and scalar multiplication.

The number of elements of S is greater than the dimension of V.

Every element of S is a linear combination of other elements of S.

There is a way to write V as a linear combination of elements of S other than the zero combination.

At least two of the elements of S are the same.

The span of S has smaller dimension than the dimension of V.

S depends on a linear transformation.

18. Let V be a vector space, and let S be a subset of V. What does it mean when we say that S is linearly independent? *

1 point

All the elements of S are distinct from each other.

The number of elements of S is less than or equal to the dimension of V.

S is a basis.

The only way to write 0 as a linear combination of elements of S is the zero combination (where one takes zero multiples of each element of S).

S has nullity zero.

S is closed under both addition and scalar multiplication.

Every element in V is a linear combination of elements in S.

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