Let $V$ be set of all pairs  $\left(x,\ y\right)$  of real numbers and let  $F$  be the field of real numbers, Define   $\left(x,\ y\right)+\left(x_1,\ y_1\right)=\left(x+x_1,\ y+y_1\right)$  and  $c\left(x,\ y\right)=\left(cx,\ y\right)$  , then

•  $V$ is a vector space over the field of real numbers.

•  $V$ is not a vector space over the field of real numbers
  

• cannot be determined

•  $V\ =\left\{\left(x,\ y\right)\ :\ x\ ,\ y\ are\ real\ numbers\right\}$  

Let  $V$  be a vector space over the field  $F$  . Let  $W$  be a subset of  $V$  . Then  $W$  is a subspace of  $V$  over the field  $F$  if  

•  $W$  is a vector space over the field  $F$  
  

•  $\forall\ x\ ,\ \ y\ \in W,\ x+y\in W\ and\ xy\in W$  

•  $\forall\ x\ ,\ \ y\ \in W\ and\ c\in F,\ x+y\in W\ and\ cy\in W$  

•  $\forall\ x\ ,\ \ y\ \in W\ and\ c\in F\ ,\ \ cx+y\in W\$  

Let  $V$ be a vector space over the field  $F$  . Let  $W_1\ ,\ W_2\ ,\ W_3$  are the subspaces of  $V$  . Then 

•  $W_1\cup\left(W_2\cap W_3\right)$  is a subspace of  $V$  

•  $W_1\cap\left(W_2\cap W_3\right)$  is a subspace of  $V$  

•  $W_1\ \cup W_2$  is a subspace of  $V$  

•  $W_2\ \cap W_3$  is a subspace of  $V$  

Let $V$ be the real vector space of all functions  $f$  from  $R\ into\ R\$  . Which of the following sets of functions are subspaces of  $V$  ? 

• all  $f$  such that  $f\left(x^2\right)=\ f\left(x\right)^2$  

• all  $f$  such that  $f\left(0\right)=f\left(1\right)$  

• all  $f$  such that  $f\left(-1\right)=0$  

• all  $f$   such that  $f\left(3\right)=1+f\left(-5\right)$  

If  $W_1\ and\ W_2$  are the finite dimensional subspaces of vector space  $V$  , then

•  $\dim\left(W_1+W_2\right)=\dim W_1+\dim\ W_2$  
  

•  $\dim\left(W_1+W_2\right)=\dim W_1+\dim\ W_2-\dim\left(W_1\cap W_2\right)$  
  

•  $\dim\left(W_1+W_2\right)+\dim\left(W_1\cap W_2\right)=\dim W_1+\dim\ W_2$  

•  $\dim\left(W_1+W_2\right)=\dim\left(W_1\cap W_2\right)$  

Let us consider $\left(1,1,2,4\right),\left(2,-1,-5,2\right),\left(1,-1,-4,0\right)\ and\ \left(2,1,1,6\right)$ vectors in  $R^4$  , then  

• they are linearly depenent

• they are linearly independent

• cannot be determined

• none of the above

Let  $V=R^3$  be a vector space over field  $R$  . Then

•  $\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,0\right)\right\}$  is a basis of  $R^3$  

•  $\left\{\left(1,0,0\right),\left(1,1,1\right),\left(0,0,2\right)\right\}$   is a basis of  $R^3$  

•  $\left\{\left(1,0,0\right),\left(0,1,0\right),\left(0,0,1\right),\left(2,3,4\right)\right\}$   is a basis of  $R^3$  

•  $\left\{\left(1,2,3\right),\left(4,5,6\right),\left(7,8,9\right)\right\}$   is a basis of  $R^3$  

Let $S$ be a linearly independent set in a vector space  $V$  .  Then which one of the following is/are true for  $S$  

• Linear span of  $S$  generates  $V$  

• For  $T\subseteq S$  ,   $T$  is also Linearly Independent.

• For  $T\subseteq S$  ,   $T$  is  Linearly dependent

• For  $S\subseteq T$  ,   $T$  is also Linearly Independent.

The value of $k$ so that vectors  $\left(1,-1,3\right),\left(1,2,-2\right),\left(k,0,1\right)$  are  linearly independent. 

• 5/4
  

• 1/4

• 3/4

• 7/4

The basis and dimension of the solution space of the homogeneous equations $x+3y+2z=0$ ,  $x+5y+z=0$  , $3x+5y+8z=0$  is  

•  $\left\{\left(-7,1,2\right)\right\}\ and\ 1$  

•  $\left\{\left(1,1,2\right)\right\}\ and\ 2$  

•  $\left\{\left(0,0,0\right)\right\}\ and\ 3$  

•  $\left\{\left(2,2,2\right)\right\}\ and\ \ 0$