Vector space

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Let $T:\ R^2\rightarrow R^3$  be a linear transformation defined by  $T\left(x,y\right)=\left(x-y,y-x,-x\right)$  , then nullity of  $T$  is 

•  $\left\{\left(1,1\right)\right\}$ 

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

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

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

The linear transformation $T:\ R^2\rightarrow R^2$  such that  $T\left(1,0\right)=\left(1,1\right)\ and\ T\left(0,1\right)=\left(-1,2\right)$  is given by  

•  $T\left(x,y\right)=\left(x-y,\ x+y\right)$  

•  $T\left(x,y\right)=\left(x+y,\ x+2y\right)$  

•  $T\left(x,y\right)=\left(x-y,\ y\right)$  

•  $T\left(x,y\right)=\left(x-y,\ x+2y\right)$  

Is null set a vector space?

• Yes

• No

Which of the following is not a meaningful expression between vectors

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\$  

In a vector space V(F), (V,+) is an......

• Group

• Monoid

• Abelian Group

• None of these

The vectors (1,-1,1), (0,1,-1), (1,0,1) in R^3 is linearly independent .

• True

• False