Verify that the mapping T : R4 -> R2 defined by
T (x1; x2; x3; x4) = (x3 - x2; 5x1 + 3x4)
is a linear transformation.
Find a basis for Im (T ) and Ker (T ).
Showing linearity:
Choose 2 vectors in $\displaystyle v_1,v_2\in \mathbb{R}^4$.
$\displaystyle v_1 = (a_1,a_2,a_3,a_4)$
$\displaystyle v_2 = (b_1,b_2,b_3,b_4)$
Choose $\displaystyle \lambda_1,\lambda_2\in \mathbb{R}$
We want to show that $\displaystyle T(\lambda_1v_1+\lambda_2v_2)= \lambda_1T(v_1)+\lambda_2T(v_2)$:
Can you do that? It's just writing out.
Observe that $\displaystyle v\in$ ker$\displaystyle (T) \Leftrightarrow T(v)= (x_3-x_2,5x_1+3x_4) = (0, 0)$. Hence v satisfies $\displaystyle x_3= x_2, x_4 = -\frac{5}{3}x_1$.
Find two independant vectors in $\displaystyle v_1,v_2\in \mathbb{R}^4$ that satisfy these relations and you have a basis for ker(T).
Find 2 more independant vectors: $\displaystyle v_3,v_4\in \mathbb{R}^4$ and you have a basis for Im(T)
(Dim(Im(T))+ Dim(ker(T)) = 4).