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

• A) Infinite-dimensional
• B) Finite-dimensional
• C)
• D)

If A and B are row equivalent matrices, then
(a) A given set of column vectors of A is ¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬_______________if and only if the corresponding column vectors of B are __________________.

• A) Linearly independent, linearly independent
• B) Linearly independent, linearly dependent
• C) Linearly dependent, linearly dependent
• D) Linearly dependent, linearly independent

The rank of a 3 x 3 matrix C (= AB), found by multiplying a non-zero column matrix A of size 3 x 1 and a non-zero row matrix B of size 1 x 3, is

• A) 3
• B) 1
• C) 0
• D) 2

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) $n-1$ vectors
• C) $n+2$ vectors
• D) $n$ vectors

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

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

If two matrices A and B are row equivalent, then their row spaces are the.

• A) Different
• B) Same
• C)
• D)

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

• A) $ColA$
• B) $RowA$
• C)
• D)

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

• A) $m-n$
• B) 0
• C) $mn$*
• D) $m+n$

The eigenvalues of a 4 by 4 matrix [A] are given as 2, -3, 1, 3 and 7. The $|det(A)|$ then is

• A) cannot be determined
• B) 546
• C) 25
• D) 530

If two matrices A and B are row equivalent, then their row spaces are the same.

• A) True
• B) False
• C)
• D)

Cylindrical coordinate system is also called as___________

• A) Spherical coordinate system.
• B) Rectangular coordinate system.
• C) Space coordinate system.
• D) Circular coordinate system.

If A is any matrix, then rank (A)=rank(A^t)

• A) False
• B) True
• 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) $${[p]_s} = (5,3,4)$$

The act of changing A into P^-1 AP is called a similarity transformation

• A) False
• B) True
• C)
• D)

Every square matrix A has at least one ----------.

• A) Critical point
• B) Fixed point
• C)
• D)

What is eigen value?

• A) A vector obtained from the coordinates
• B) A matrix determined from the algebraic equations
• C) It is the inverse of the transform
• D) A scalar associated with a given linear transformation

The number of parameters in the solution set of Ax = 0 if A is a 4× 4 matrix of rank 0 then nullity (A) =?

• A) 0
• B) 4
• C)
• D)

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)

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

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