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.

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) True
• B) False
• C)
• D)

If u and v are non zero vectors in either R^2 or R^3 then by the law of cosines ||u-vII^2=---------

• A) ||u-v||^2=||u||^2+||v||^2-2||u||||v||cos \theta
• B) ||u-v||^2=||u||^2+||v||^2+2||u||||v||cos \theta
• C)
• D)

Let W be a subspace of R^n, then each y in R^n can be written uniquely in the form

• A) y=\widehat y -z
• B) y=\widehat y /z
• C) none of the above
• D) y=\widehat y +z

{\text{The }}\_\_\_\_\_ ~vector~ is ~orthogonal~ to ~every ~vector ~in~ {R^n}

• A) none of these
• B) zero
• C) unit
• D) normalized

If there is a vector u=(1, -1, 2) then ||u|| is------

• A) \sqrt{6}
• B) 0
• C) 2
• D) \sqrt {7}

If there is a vector v=(2, 1, 0) then ||v|| is ------

• A) 3
• B) 2
• C) \sqrt{5}
• D) 0

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 \{u_{1}, u_{2}, …, u_{p}\} is an orthonormal basis for a subspace W of R^n, then ...

• A) Proj_{w}y=(y.u_{1})u_{1}+(y.u_{2})u_{2}+…+(y.u_{p})u_{p}
• B) Proj_{w}y=(y.u_{1})u_{p}+(y.u_{2})u_{p}+…+(y.u_{p})u_{p}
• C)
• D)

\[{\text{The }}\_\_\_\_\_ ~vector~ is ~orthogonal~ to ~every ~vector ~in~ {R^n}\]

• A) unit
• B) zero
• C) none of these
• D) normalized

Let W be a subspace of R^n, y be any vector in R^n and \widehat y the orthogonal projection of y onto W. Then \widehat y is the closest point in W to y, in the sense that

• A) ||y-\widehat y ||>||y-v||
• B) ||y-\widehat y||<||y-v||
• C)
• D)

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

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

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

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

• A) Eigenfunction
• B) Eigensolution
• C)
• D)

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

• A) True
• B) False
• C)
• D)