A set of two non-zero vectors are linearly independent if and only if they are --------------- to/of each other.

• A) Equal
• B) Scalar multiple
• C)
• D)

For a given matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 2}&{ - 3}&{ - 4}&{ - 4}&6 \\\ 0&0&1&2&1&{ - 4} \\\ 0&0&0&0&1&7 \\\ 0&0&0&0&0&0 \end{array}} \right]\] which of the following are basis vectors.

• A) \[\left[ {\begin{array}{*{20}{c}} 1 \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 2} \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} 6 \\\ { - 4} \\\ 7 \\\ 0 \end{array}} \right]\]
• B) \[\left[ {\begin{array}{*{20}{c}} 1 \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 4} \\\ 1 \\\ 1 \\\ 0 \end{array}} \right]\]
• C) \[\left[ {\begin{array}{*{20}{c}} 1 \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 2} \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right]\]
• D) \[\begin{gathered} \left[ {\begin{array}{*{20}{c}} 1&{ - 2}&{ - 3}&{ - 4}&{ - 4}&6 \end{array}} \right] \hfill \\ \left[ {\begin{array}{*{20}{c}} 1&0&1&2&1&{ - 4} \end{array}} \right] \hfill \\ \left[ {\begin{array}{*{20}{c}} 0&0&0&0&1&7 \end{array}} \right] \hfill \\\ \end{gathered} \]

The only 0-dimensional subspace of R^3 is ---------

• A) Uni space
• B) Zero space
• C)
• D)

The set B={1,x,x^2,…,x^n} forms a basis for the vector space P_n of polynomial of degree

• A) \leq n
• B) None of these
• C) \geq n
• D) n

Any set of linearly independent vectors cannot contain the ----------- vector.

• A) Unit
• B) Identity
• C) zero
• D) None of these

\[If~ A ~is~ an~ m \times n ~matrix ~then\]

• A) \[rank A = \dim NulA + n\]
• B) \[rank A = \dim Nul A\]
• C) \[rank A = n\]
• D) \[rank A + \dim NulA = n\]

If Coordinates vector of the polynomial p=\[5 - 4x - 3{x^2}\] relative to the basis S = \[\{ 1,x,{x^2}\} \] for \[{p_2}\]then linear combination of the basis set S is

• A) \[{[p]_s} = (5, - 4, - 3)\]
• B) \[{[p]_s} = (5,4,3)\]
• C) \[{[p]_s} = (5,3,4)\]
• D) \[{[p]_s} = (5, - 4,3)\]

Rank of the matrix A=\left[ \begin{gathered} 0\,\,\,0\,\,\,0\,\,\,0 \hfill \\ 4\,\,\,6\,\,\,\,1\,\,\,\,0\, \hfill \\ 1\,\,\,\,\,0\,\,\,\,0\,\,\,\,0 \hfill \\\ \end{gathered} \right]

• A) 1
• B) 0
• C) 2
• D) 3

Let $S = \left\{ {{v_1},{v_2}, \ldots ,{v_n}} \right\}$ be a set in V and le $H = span\left\{ {{v_1},{v_2}, \ldots ,{v_p}} \right\}$ . Some subsets of S are basis for H if --------------

• A) $H={0}$
• B) $H \ne \left\{ 0 \right\}$
• C)
• D)

If n>m, then the set \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\} of vectors in {R^m} must be ---------

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

Let S = \left\{ {{v_1},{v_2}, \ldots ,{v_n}} \right\} be a set in V and le H = span\left\{ {{v_1},{v_2}, \ldots ,{v_p}} \right\} . Some subsets of S are basis for H if --------------

• A) H \ne \left\{ 0 \right\}
• B) H={0}
• C)
• D)

If Coordinates vector of the polynomial \[p = {a_0} + {a_1}x + {a_2}{x_2}\] Relative to the basis S = \[\{ 1,x,{x^2}\} \,\,\,\] for \[{p_2}\] then linear combination of the basis set S is

• A) \[{[p]_s} = ({a_0} + {a_1} + {a_2})\]
• B) \[{[p]_s} = [{a_0} + {a_1} + {a_2}]\]
• C)
• D)

An indexed set of vectors ${v_1,v_2,…,v_p}$ in $V$ is said to be linearly independent if the vector equation

• A) ${c_1}{v_1} + {c_2}{v_2} + \ldots + {c_p}{v_p} = 1$
• B) None of the above.
• C) ${c_1}{v_1} + {c_2}{v_2} + \ldots + {c_p}{v_p} =0 $
• D) ${c_1}{v_1} + {c_2}{v_2} + \ldots + {c_p}{v_p} \ne 0$

If Coordinates vector of the polynomial p = {a_0} + {a_1}x + {a_2}{x_2} Relative to the basis S = \{ 1,x,{x^2}\} \,\,\, for {p_2} then linear combination of the basis set S is

• A) {[p]_s} = [{a_0} + {a_1} + {a_2}]
• B) {[p]_s} = ({a_0} + {a_1} + {a_2})
• C)
• D)

\[\begin{gathered} if~ A ~is~ a~ 4 \times 5~ matrix~ with~ a~ two ~- {\text{dimentinal}}~ null space, ~ \hfill \\ what~ is~ rank ~of ~A \hfill \\\ \end{gathered} \]

• A) 6
• B) 5
• C) 7
• D) 3

If n>m, then the set $\left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}$ of vectors in ${R^m}$ must be ---------

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

If n>m, then the n vectors \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\} are --------- of an m \times n matrix A

• A) Rows
• B) Columns
• C)
• D)

In \dim \left( {{R^n}} \right) = n, the standard basis has ------- vectors

• A) n
• B) n+1
• C) mn
• D) 0

Rank of the matrix A =\left[ {\begin{array}{*{20}{c}} 0&0&0&0 \\\ 4&6&1&0 \\\ 1&0&0&0 \end{array}} \right]

• A) 2
• B) 1
• C) 0
• D) 3

In two coordinate system, {\left( {\mathop P\limits_{C \leftarrow B} } \right)^{ - 1}} is the matrix that converts

• A) C-coordinates into C-coordinates
• B) B-coordinates into C-coordinates
• C) B-coordinates into B-coordinates
• D) C-coordinates into B-coordinates