PlZ Ans the optional questions
Q1. The vector space V3(R) is of dimension:
(a) 1
(b) 3
(c) 2
(d) none of these
Q2. The zero vector in the vector space R4 is:
(a) (0, 0)
(b) (0, 0, 0)
(c) (0, 0, 0, 0)
(d) none of these
Q3. A set of vectors which contains at least one zero vector is:
(a) linearly independent
(b) linearly dependent
(c) linear space
(d) none of these
Q4. If X has a basis consisting of n elements than any other basis of X is:
(a) n elements
(b) n2 elements
(c) n–1 elements
(d) one elements
Q5. If in an norm product space V(F), the vector α is O then (α, α) =
(a) 0
(b) O
(c) 1
(d) none of these
Q6. The norm of the vector α =(1, –2, 5) is:
(a)
(b) 30
(c) 8
(d) none of these
Q7. For matrices A and B, (AB)' is:
(a) A'B'
(b) A' + B'
(c) B' A'
(d) none of these
Q8. The rank of the matrix of order mxn is:
(a) ≤ m
(b) ≤ n
(c) ≤ min(m, n)
(d) none of these
Q9. The eigen values of an orthogonal matrix are:
(a) zero
(b) imaginary
(c) real
(d) of unit modulus
Q10. A real symmetric matrix is positive definite if all its eigen values are:
(a) positive
(b) negative
(c) complex
(d) zero
Dear Student,
You are requested to ask 1 doubt at a time. Few of your questions are beyond class 11-12 syllabus such as terms like norm,basis,inner product space and eigen values,positive definite are all taught in Linear Algebra in Engineering. Still I have tried to answer your queries briefly.
Q1. B) Vn(R) has n dimensions. So, V3(R) is a 3 dimensional vector.
Q2. C) A zero vector is 0 in all directions. Hence (0,0,0,0) is correct option.
Q3. B) Any vector having atleast one zero vector is always linearly dependent as you can always form a non zero coefficient set. Eg:- for a vector V=(a,b,c,0,e,f) we can have coefficient set (0,0,0,k,0,0) where k is a non zero. Hence it is linearly dependent.
Q4. A). The basis has always n elements for Vn.
Q5. A) Inner product space can be imagined as a dot product at basic level.It has entirely different physical significance. Infact dot product is studied under inner product space.Since the vector is O , its magnitude is 0 and <v,w> is always greater than equal to 0. Equality holds iff v=w=0
Q6. A) Norm of a vector is simply equal to its magnitude.
Q7. A) It is a property of transposition of Matrices (AB)' = A'B'
Q8.C) Again it is studied vastly in Linear Algebra. In M*N matrix, Row Rank <= m and Column Rank <= n. But it can be proved that both row and column rank are equal . Hence Rank of matrix <= min(m,n).
Q9.D) Again, the proof uses the defination and properties of Eigen values. I can give you the result which says if A is orthogonal, then all eigenvalues are equal to -1 or 1.
Q10.
You are requested to ask 1 doubt at a time. Few of your questions are beyond class 11-12 syllabus such as terms like norm,basis,inner product space and eigen values,positive definite are all taught in Linear Algebra in Engineering. Still I have tried to answer your queries briefly.
Q1. B) Vn(R) has n dimensions. So, V3(R) is a 3 dimensional vector.
Q2. C) A zero vector is 0 in all directions. Hence (0,0,0,0) is correct option.
Q3. B) Any vector having atleast one zero vector is always linearly dependent as you can always form a non zero coefficient set. Eg:- for a vector V=(a,b,c,0,e,f) we can have coefficient set (0,0,0,k,0,0) where k is a non zero. Hence it is linearly dependent.
Q4. A). The basis has always n elements for Vn.
Q5. A) Inner product space can be imagined as a dot product at basic level.It has entirely different physical significance. Infact dot product is studied under inner product space.Since the vector is O , its magnitude is 0 and <v,w> is always greater than equal to 0. Equality holds iff v=w=0
Q6. A) Norm of a vector is simply equal to its magnitude.
Q7. A) It is a property of transposition of Matrices (AB)' = A'B'
Q8.C) Again it is studied vastly in Linear Algebra. In M*N matrix, Row Rank <= m and Column Rank <= n. But it can be proved that both row and column rank are equal . Hence Rank of matrix <= min(m,n).
Q9.D) Again, the proof uses the defination and properties of Eigen values. I can give you the result which says if A is orthogonal, then all eigenvalues are equal to -1 or 1.
Q10.