MCQ Bank
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)
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.