Linear Algebra MCQs

1

Advanced Mathematics • Linear Algebra

Consider the following real-valued maps defined over the vector space of 2 x 2 real-valued matrices. Which one of the following is not linear? Consider the following real-valued maps defined over the vector space of 2 x 2 real-valued matrices. Which one of the following is not linear?

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A. T(A) = tr(A) T(A) = tr(A) B. T(A) = a_11 T(A) = a_11 C. T(A) = A^t T(A) = A^t D. T(A) = det(A) T(A) = det(A)

सही उत्तर: D Correct Answer: D

The determinant is not a linear map because, in general, det(A + B) is not equal to det(A) + det(B), and det(cA) is not equal to c det(A).

2

Advanced Mathematics • Linear Algebra

A is a square matrix with each row sum equal to 1. Which statement regarding the eigenvalues of A is always true? A is a square matrix with each row sum equal to 1. Which statement regarding the eigenvalues of A is always true?

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A. 0 is an eigenvalue of A. 0 is an eigenvalue of A. B. 1 is an eigenvalue of A. 1 is an eigenvalue of A. C. Both 0 and 1 are eigenvalues of A. Both 0 and 1 are eigenvalues of A. D. No statement can be made about an eigenvalue of A from the given information. No statement can be made about an eigenvalue of A from the given information.

सही उत्तर: B Correct Answer: B

If e is the column vector whose entries are all 1, then Ae = e because every row sum is 1. Therefore 1 is an eigenvalue.

3

Advanced Mathematics • Linear Algebra

Which one of the following statements is false? Which one of the following statements is false?

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A. C[0,1] denotes the space of all continuous functions. T:C[0,1] to R defined as T(f) = 2f(1) + integral from 0 to 1 of f(s)ds is a linear functional. C[0,1] denotes the space of all continuous functions. T:C[0,1] to R defined as T(f) = 2f(1) + integral from 0 to 1 of f(s)ds is a linear functional. B. M_2x3(R), the space of all real-valued 2 x 3 matrices, is isomorphic to R^5. M_2x3(R), the space of all real-valued 2 x 3 matrices, is isomorphic to R^5. C. Let A,B:V to V be linear and I be the identity map. Then AB = I implies that A and B are invertible. Let A,B:V to V be linear and I be the identity map. Then AB = I implies that A and B are invertible. D. A linear transformation T is invertible if and only if T is one-one and onto. A linear transformation T is invertible if and only if T is one-one and onto.

सही उत्तर: B Correct Answer: B

The vector space of 2 x 3 real matrices has dimension 6, so it is isomorphic to R^6, not R^5.

4

Advanced Mathematics • Linear Algebra

Obtain the dimension of the dual space for {(x,y,z): x + 2y + 3z = 0}. Obtain the dimension of the dual space for {(x,y,z): x + 2y + 3z = 0}.

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A. 3 3 B. 2 2 C. 4 4 D. 1 1

सही उत्तर: B Correct Answer: B

The given set is the null space of one nonzero linear equation in R^3, so it has dimension 2. A finite-dimensional vector space and its dual have the same dimension.

5

Advanced Mathematics • Linear Algebra

Which one of the following statements is false? Which one of the following statements is false?

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A. For any vector space, a basis set gives a minimal spanning set. For any vector space, a basis set gives a minimal spanning set. B. A basis set for a vector space is not unique. A basis set for a vector space is not unique. C. Any set of vectors containing the null vector is linearly dependent. Any set of vectors containing the null vector is linearly dependent. D. For any linear transformation T:V to V and any basis {v1,...,vk} of V, the set {T(vi): 1 <= i <= k} also gives a basis set. For any linear transformation T:V to V and any basis {v1,...,vk} of V, the set {T(vi): 1 <= i <= k} also gives a basis set.

सही उत्तर: D Correct Answer: D

The image of a basis under an arbitrary linear transformation need not be a basis; this is guaranteed only when the transformation is invertible.

6

Advanced Mathematics • Linear Algebra

Which one of the following is NOT a subspace? Which one of the following is NOT a subspace?

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A. {(x,2x): x belongs to R} over R {(x,2x): x belongs to R} over R B. Q over Q Q over Q C. {(x,y): xy = 0} over R {(x,y): xy = 0} over R D. {(x,y): 2x + 3y = 0} over R {(x,y): 2x + 3y = 0} over R

सही उत्तर: C Correct Answer: C

The set xy = 0 is the union of the coordinate axes and is not closed under addition; for example, (1,0) and (0,1) belong to it but (1,1) does not.

7

Advanced Mathematics • Linear Algebra

What is the dimension of the vector space of all 2 x 2 symmetric real-valued matrices? What is the dimension of the vector space of all 2 x 2 symmetric real-valued matrices?

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A. 3 3 B. 1 1 C. 2 2 D. 4 4

सही उत्तर: A Correct Answer: A

A symmetric 2 x 2 matrix has the form [[a,b],[b,c]], determined by three independent real parameters. Its dimension is 3.

8

Advanced Mathematics • Linear Algebra

A is an n x n non-singular matrix. Which one of the following statements is true? A is an n x n non-singular matrix. Which one of the following statements is true?

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A. The eigenvectors of A are orthogonal to each other. The eigenvectors of A are orthogonal to each other. B. All the eigenvalues are positive. All the eigenvalues are positive. C. 0 cannot be an eigenvalue of A. 0 cannot be an eigenvalue of A. D. tr(A) = sum from i=1 to n of a_ii is not equal to 0. tr(A) = sum from i=1 to n of a_ii is not equal to 0.

सही उत्तर: C Correct Answer: C

A matrix is non-singular exactly when its determinant is nonzero. Since the determinant is the product of its eigenvalues, zero cannot be an eigenvalue.

9

Advanced Mathematics • Linear Algebra

Consider the vector space of all polynomials with real coefficients and degree bounded by 2, and define the linear functional T as T(f)=f(0). Find the dimension of the null space of T. Consider the vector space of all polynomials with real coefficients and degree bounded by 2, and define the linear functional T as T(f)=f(0). Find the dimension of the null space of T.

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A. 2 2 B. 4 4 C. 1 1 D. 3 3

सही उत्तर: A Correct Answer: A

The space has basis {1,x,x^2} and dimension 3. T has rank 1, so rank-nullity gives nullity 3-1=2.

10

Advanced Mathematics • Linear Algebra

For any subset U of a vector space V, let U-perpendicular denote the set of vectors orthogonal to every vector in U. Find (U-perpendicular)-perpendicular, where U={(2,0,0),(1,0,3)}. For any subset U of a vector space V, let U-perpendicular denote the set of vectors orthogonal to every vector in U. Find (U-perpendicular)-perpendicular, where U={(2,0,0),(1,0,3)}.

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A. R^3 R^3 B. {(2,0,0),(1,0,3)} {(2,0,0),(1,0,3)} C. {(x,y,z): y=0} {(x,y,z): y=0} D. {(0,0,0)} {(0,0,0)}

सही उत्तर: C Correct Answer: C

The two vectors are independent and span the xz-plane y=0. In finite-dimensional inner-product spaces, (U-perpendicular)-perpendicular equals the span of U.

Linear Algebra MCQs

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