Answer to Question #143185 in Linear Algebra for Sourav Mondal

Question

Let V be the subspace of R3 spanned by

{(1, 1, 0), (1, 1, 1)} and T : V —> V be defined by T(x1, x2, x3) = (0, x1, x2).

Find the kernel of T.

Answer 1

Consider the spanning set of $V$ . It means that any vectors in $V$ can be written as a linear combination of $\{(1,1,0),(1,1,1)\}.$ That is, for any $(x_1,x_2,x_3) \in V,$

$(x_1,x_2,x_3)=\lambda_1(1,1,0)+\lambda_2(1,1,1), \lambda_1,\lambda_2 \in \R\\ (x_1,x_2,x_3)=(\lambda_1+\lambda_2,\lambda_1+\lambda_2,\lambda_2)$

The Kernel of T is defined as the set;

$Ker(T)=\{T(x_1,x_2,x_3)=(0,0,0)|(x_1,x_2,x_3)\in V\}.$

To get $Ker(T)$ ;

$T(x_1,x_2,x_3)=(0,0,0)\\(0,x_1,x_2)=(0,0,0)\\ \implies x_1=0, x_2=0$ whereby, $x_3$ remains unchanged. That is, $x_3\in \R$ .

$\implies \lambda_1=-\lambda_2$ , $x_3=\lambda_2$

Therefore, the $Ker(T):=\{(0,0,x_3)|x_3\in \R\}$ and its basis is $(0,0,1)$ .