A set of single non-zero vector is

• A) None of the above
• B) Linearly Independent
• C) Basis
• D) Linearly Dependent

A single non-zero vector always forms a linearly -------------- set.

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

The vectors spaces F(-\infty, +\infty) and C(-\infty, +\infty) are ----------

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

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) Columns
• B) Rows
• C)
• D)

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, - 4,3)
• D) {[p]_s} = (5,3,4)

If a vector space V has a basis of n vectors then every basis of V must consists of exactly --------

• A) n-1 vectors
• B) n vectors
• C) n+2 vectors
• D) n+1 vectors

If a vector space V has a basis B={b_1, b_2, …, b_n} then any set in V, containing more than n vectors, must be -----------

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

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) 2
• C) 3
• D) 0

Dim(M_{m\times n})=-------

• A) m-n
• B) m+n
• C) mn*
• D) 0

The dimension of the zero vector space \left\{ 0 \right\} is defined to be

• A) 1
• B) None of these
• C) 0
• D) \infty

A vector space V with a basis B containing n vectors is isomorphic to R^n i.e., there exist a ------------- linear transformation from V to R^n

• A) One-to-one
• B) Onto
• 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) 7
• B) 5
• C) 3
• D) 6

The set of unit vectors {i,j,k}form a basis of

• A) R^2
• B) R
• C) R^4
• D) R^3

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}} { - 2} \\\ 0 \\\ 0 \\\ 0 \end{array}} \right], \left[ {\begin{array}{*{20}{c}} { - 3} \\\ 1 \\\ 0 \\\ 0 \end{array}} \right]
• C) \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]
• 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}

If B=/{b_{1}, b_{2}, …, b_{n}/} and E is the standard basis /{e_{1}, e_{2}, …, e_{n}/} in R^{n}, then

• A) [b_{1}]_{E}=0
• B) [b_{1}]_{E}=b_{1}
• C)
• D)

If A is any matrix, then Rank(A) = Rank({A^T})

• A) false
• B) True
• C)
• D)

The pivot columns of a matrix A form a basis for -----------

• A) ColA
• B) RowA
• C)
• D)

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

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

Rank of the matrix A=

• A) 1
• B) 3
• C) 2
• D) 0