Corresponding to highest Eigen value the eigenvector is \[\left( {\begin{array}{*{20}{c}} 1&6&1 \\\ 1&2&0 \\\ 0&0&3 \end{array}} \right)\]

• A) \[\left[ {\begin{array}{*{20}{c}} 1 \\\ 2 \\\ 0 \end{array}} \right]\]
• B) \[\left[ {\begin{array}{*{20}{c}} 0 \\\ 1 \\\ 2 \end{array}} \right]\]
• C) \[\left[ {\begin{array}{*{20}{c}} 2 \\\ 0 \\\ 1 \end{array}} \right]\]
• D) \[\left[ {\begin{array}{*{20}{c}} 2 \\\ 1 \\\ 0 \end{array}} \right]\]

\[An{\text{_____}} matrix~with~n~distinct~eigenvalues~is~diagonlizable\]

• A) None of these
• B) Null
• C) \[m \times n\]
• D) \[n \times n\]

$\phi $ is the angle between the positive x-axis and the ray

• A) From (a,b) through (0,0)
• B) From (0,0) through (a,b)
• C) None of these
• D) From (1,1) through (0,0)

If A is m\cross n and b is in R^n, a least square solution of Ax=b is an \widehat x in R^n such that

• A) ||b-A\widehat x ||<=||b-Ax|| \forall x \in R^n
• B) ||b-A\widehat x||>=||b-Ax|| \forall x \in R^n
• C)
• D)

The zero vector is orthogonal to every vector in R^n

• A) True
• B) False
• C)
• D)

Let W is a finite dimensional subspace of an inner product space V and y is any vector in V. The best approximation to y from W is then

• A) Proj^{y}_{w}
• B) Proj^{w}_{y}
• C)
• D)

Each pair of eigenvalue and its corresponding eigenvector provides a solution of the equation x’=Ax which is called ---------------- of the differential equation.

• A) Eigen functions
• B) Eigen vectors
• C)
• D)

A m ×n matrix U has orthonormal columns if and only if {U^t}U = I

• A) True
• B) False
• C)
• D)

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)

Let W be a subspace of {R^n} and \{ {u_1},{u_2},...,{u_p}\} is any orthogonal basis of W, then \mathop y\limits^\^ = {c_1}{u_1} + {c_2}{u_2} + ... + {c_n}{u_n} Where

• A) {c_j} = \frac{{y.{u_j}}}{{u_j}}
• B) None of these
• C) {c_j} = \frac{{y.{u_j}}}{y}
• D) {c_j} = \frac{{y.{u_j}}}{{{u_j}.{u_j}}}

If a vector x_0 is specified, then the initial value problem is to construct the unique function x such that --------

• A) x’=Ax and x(0)=x_0
• B) x’=Ax’ and x(0)=x_0
• C)
• D)

How many subspaces does R^2 have?

• A) None of these
• B) Infinitely many
• C) 0 and R^2
• D) 0, R×0,0×R,R^2

Let u, v and w be vectors in {R^n}, then

• A) (u+v).w=u.w+v.w
• B) (u+v).w=u.w+v.w-2u.w
• C) (u+v).w=u.w-v.w
• D) None of the above

Two vectors u and v in R^n are orthogonal if ------

• A) u.v \neq 0
• B) u.v=0
• C)
• D)

For u and v vectors in R^n, thw distance between u and v written as dist(u, v) is the length of the vector------

• A) u-v
• B) v-u
• C)
• D)

Suppose that real solutions y_1 and y_2 of x’=Ax, form a basis for the two-dimensional real vector space if y_1 and y_2 are ……..

• A) Linearly Independent
• B) Linearly dependent
• C)
• D)

Let s = \{ {u_1} + {u_2} + ......{u_p}\} be a basis for a subspace W of {R^n} , is also an orthogonal basis if S is an orthogonal set.

• A) True
• B) False
• C)
• D)

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) True
• B) False
• C)
• D)

If u and v are vectors in R^n, then we regard u and v as n \times 1 matrices. The matrix product u^{t}v is a ----------- matrix.

• A) 1 \times 1
• B) n \times n
• C) 1 \times n
• D) n \times 1

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 block upper triangular matrix.
• C) It is a block lower triangular matrix.
• D) It is a Null matrix.