A transformation $T:{R^n} \to {R^m}$ Is a rule that assigns to each vector x in ${R^n}$ , an image vector T(x) in ${R^m}$. The set ${R^n}$ is called the__________ , and ${R^m}$ is called the _______.

• A) domain of T, co-domain of T
• B) co-domain of T, domain of T
• C)
• D)

Which one of the following is a null matrix?

• A) $$\left( {\begin{array}{*{20}{c}} 0&0&0 \\\\\\\\\ 0&0&0 \\\\\\\\\ 0&0&0 \end{array}} \right)$$
• B) $$\left( {\begin{array}{*{20}{c}} 1&0&0 \\\\\\\\\ 2&0&0 \\\\\\\\\ 0&0&1 \end{array}} \right)$$
• C)
• D)

A null space is a vector space.

• A) False
• B) True
• C)
• D)

$${\text{Vectors}} \left( {\begin{array}{*{20}{c}} { - 2} \\\ 3 \end{array}} \right){\text{ and}} \left( {\begin{array}{*{20}{c}} { - 66} \\\ {99} \end{array}} \right){\text{ are linearly }} - - - - - .$$

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

$${\text{If }}A = \left( {\begin{array}{*{20}{c}} 1&2 \\\ 3&6 \end{array}} \right)\mathop \sim \limits^{{\text{Row Equivalent}}} \left( {\begin{array}{*{20}{c}} 1&2 \\\ 0&0 \end{array}} \right), {\text{then the Columns of}} ~{\text{A}} ~{\text{are}} ~{\text{Linearly}} {\text{ - - - - - - - }}$$

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

$${\text{Vector}} \left( {\begin{array}{*{20}{c}} { - 1} \\\ 0 \end{array}} \right) {\text{is Linearly}} - - - - - - .$$

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

The two vectors $(-2,1)$ and $(1,2)$ are

• A) Perpendicular to each other
• B) Pointing in the opposite direction of each other
• C) Linearly dependent of each other
• D) None of the above

What is the maximum possible number of pivots in a 4x6 matrix ?

• A) 6
• B) 8
• C) 4
• D) 10

If A= transpose of A( where A is a square matrix ), then which of the following is the most appropriate option for A?

• A) A is invertible matrix.
• B) A is a scalar matrix.
• C) A is symmetric matrix.
• D) A is singular matrix.

A system of linear equations is said to be homogeneous if it can be written in the form ______________.

• A) AX = 0
• B) AX = B
• C)
• D)

$${\text{Vectors}} \left( {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}} \right) {\text{and}} \left( {\begin{array}{*{20}{c}} 0 \\\ 1 \end{array}} \right) {\text{are linearly}} - - - - - .$$

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

Let A be the matrix of order 2X3 and B be the matrix of order 3X5,then which of the following is the order of the matrix AB?

• A) 2x5
• B) 3X5
• C) 3X3
• D) 2X3

$${\text{Since vector}}\left( {\begin{array}{*{20}{c}} 2 \\\ 3 \end{array}} \right) {\text{lies in the span}} \left\{ {\left( {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 0 \\\ 1 \end{array}} \right)} \right\} {\text{then the vectors }}\left( {\begin{array}{*{20}{c}} 2 \\\ 3 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 1 \\\ 0 \end{array}} \right) {\text{and}} \left( {\begin{array}{*{20}{c}} 0 \\\ 1 \end{array}} \right) {\text{are Linearly }} - - - -$$

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

Any set $$\{ {v_1},{v_2},...,{v_p}\} \in {R^n}$$ is linearly dependent if

• A) p<n
• B) p>n
• C)
• D)

$${\text{Set}} \left\{ {\left( {\begin{array}{*{20}{c}} 1 \\\ 2 \end{array}} \right),\left( {\begin{array}{*{20}{c}} 0 \\\ 0 \end{array}} \right)} \right\} {\text{is Linearly - - - - - in}}~ {\mathbb{R}^2}.$$

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

$$\eqalign{ & If A = {A^t} ( Where A is a square matrix), then which of the following \cr & is the most appropriate option for A \cr}$$

• A) A is a scalar matrix.
• B) A is invertible matrix.
• C) A is singular matrix.
• D) A is symetric matrix.

A set of two vectors $$\{ {v_1},{v_2}\}$$ is linearly dependent if and only if one of the vector is

• A) Additive inverse of the other
• B) Multiple of the other
• C)
• D)

The columns of a matrix A are ------------------- if and only if the equation $Ax=0$ has only the trivial solution.

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

Which of the following is true for the partitioned matrices $Y = [\,Q\,\,\,\,\,\,\,\,R\,]\,\,\,{\rm{and}}\,\,\,Z = [\,U\,\,\,\,\,\,\,\,V\,]\,,$ where sub-matrices $Q$ and $R$ have the same sizes as $U$ and $V$ respectively?

• A) $$Y + Z = [\,Q + U\,\,\,\,\,\,\,R + V\,]\,$$
• B) $$Y + Z = [\,QU\,\,\,\,\,\,\,RV\,]\,$$
• C) $$Y + Z = \left[ \begin{array}{l} \,QU\\\RV \end{array} \right]$$
• D) $$Y + Z = \left[ \begin{array}{l} \,Q + U\\\R + V\, \end{array} \right]$$

Which of the following is true for the speed of convergence of both the Jacobi and Gauss-Seidel sequences?

• A) It depends on how much the diagonal entries dominate the corresponding row sums.
• B) It depends on how much the diagonal entries dominate the corresponding column sums.
• C) It depends on how much the diagonal entries dominate the preceding column sums.
• D) It depends on how much the diagonal entries dominate the preceding row sums.