Let W be a subspace of ${R^n}$ and $\{ {u_1},{u_2},...,{u_p}\} $ is an orthonormal basis, then

• A) $\Pr o{j_w}y = (y.y){u_1} + (y.y){u_2} + ... + (y.y){u_p}$
• B) $\Pr o{j_w}y = (y.{u_1}){u_1} + (y.{u_2}){u_2} + ... + (y.{u_p}){u_p}$
• C)
• D)

Each pair of eigenvalue and its corresponding eigenvectors provides a solution of the equation \[x' = Ax\] which is called

• A) vector space
• B) solution matrix
• C) dynamical system
• D) eigenfunctions

If B=\{ {v_1},{v_2},{v_3}\} is an orthogonal set of vectors with respect to an inner product on a vector space V, then the set B

• A) Spans the vector space v
• B) Is linearly independent.
• C) Linearly dependent.
• D) Is an orthonormal basis for v

If \[s = \{ {u_1} + {u_2} + ......{u_p}\} \] is an orthogonal basis for a subspace W of \[{R^n}\] Then each y in W can be uniquely expressed as a linear combination of \[{u_1} + {u_2} + ......{u_p}\] That is\[{c_1}{u_1} + {c_2}{u_2} + ......{c_p}{u_p}\] Where\[\frac{{y.{u_j}}}{{{u_j}.{u_j}}}\]

• A) False
• B) True
• C)
• D)

For u, v vectors in {R^n}, the distance between u and v is written as

• A) dist(u,v) = ||u - u||
• B) dist(u,v) = ||u - v||
• C) dist(u,v) = ||v-v||
• D) None of these

Which of the following is true for the matrix M = \left[ \begin{array}{l} \,{M_{\,11}}\,\,\,\,\,\,\,{M_{\,12}}\,\,\,\,\,\,\,{M_{\,13}}\\\ O\,\,\,\,\,\,\,\,\,\,\,\,{M_{\,22}}\,\,\,\,\,\,\,{M_{\,23}}\\\ O\,\,\,\,\,\,\,\,\,\,\,\,\,O\,\,\,\,\,\,\,\,\,\,\,\,{M_{\,33}}\, \end{array} \right]\,; where {M_{\,\,11}} , {M_{\,22}} and {M_{\,33}} are square sub-matrices , and O is a zero sub-matrix ?

• A) It is diagonal-constant matrix.
• B) It is a Null matrix.
• C) It is a block lower triangular matrix.
• D) It is a block upper triangular matrix.

If B=\[\{ {v_1},{v_2},{v_3}\} \] is an orthogonal set of vectors with respect to an inner product on a vector space V, then the set B

• A) Is linearly independent.
• B) Is an orthonormal basis for v
• C) Spans the vector space v
• D) Linearly dependent.

A matrix [A]_{n \times n has both positive and negative eigenvalues so in this case origin behaves as a --------------

• A) Critical point
• B) Saddle point
• C)
• D)

Let \[s = \{ {u_1} + {u_2} + ......{u_p}\} \] be the set of non-zero vectors in \[{R^n}\] is said to be an orthogonal set if all vectors in S are mutually orthogonal. That is O∉ S and \[{u_i}.{u_j} = 0\,\] ∀ i≠ j, i,j=1,2,……..p.

• A) False
• B) True
• C)
• D)

The norm of v is the non-negative scalar||v|| defined by

• A) ||v||=\sqrt{v_{1}^{2}+v_{2}^{2}+...v_{n}^{2}}
• B) ||v||=v_{1}^{2}+v_{2}^{2}+...v_{n}^{2}
• C)
• D)

If A and B are row equivalent matrices, then (a) A given set of column vectors of A is ¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬_______________if and only if the corresponding column vectors of B are __________________.

• A) Linearly independent, linearly independent
• B) Linearly independent, linearly dependent
• C) Linearly dependent, linearly dependent
• D) Linearly dependent, linearly independent

Eigenvector(s) of the matrix \[\left[ {\begin{array}{*{20}{c}} 0&0&a \\ 0&0&0 \\ 0&0&0 \end{array}} \right]\] is (are):

• A) (0, a, 0)
• B) None of the above
• C) (0, 0, 1)
• D) (0, 0, a)

Which of the following best describes the simplex method?

• A) An approach for solving integer programming problems.
• B) A method for solving quadratic programming problems.
• C) A deterministic algorithm for solving linear programming problems.
• D) A probabilistic algorithm for solving nonlinear programming problems.

While solving a LP by Simplex method, which of the following corresponds to corner points of the feasible region in Graphical method?

• A) basic feasible solution
• B) non-negative solutions
• C) non-basic feasible solution
• D) non-feasible solution

While solving an LP problem by the Simplex method, in the standard table, the element at the intersection of key column and key row is called ------------- element.

• A) Slack
• B) Entering
• C) Leaving
• D) Pivot

The variants of the simplex method cannot be identified through numerical examples.

• A) False
• B) True
• C)
• D)

After converting constraints into the respective Standard equalities, we have an LP problem of ‘4’ equations in ‘6’ variables, then in a given iteration, how many Basic feasible VARIABLES are possible to exists?

• A) 2
• B) 4
• C) 10
• D) 6

Which method is also known as the Charnes method of penalty in the artificial variable technique?

• A) Penalty Method
• B) Big Method
• C) Two Phase Method
• D) Slack Variable Method

While solving a Linear Programming problem by Simplex Method, an inequality says:‘2x-3y>5’ is transformed into strict equality of the form---------, where ‘s>=0’

• A) 2x-3y +s + A=5
• B) 2x-3y –s +A =5
• C) 2x-3y –s =5
• D) 2x-3y +s =5

The straight line associated with the constraint:2x+3y <12, will meet y-axis at a distance of-------- units from origin.

• A) 4
• B) 6
• C) 3
• D) 2