Find a linear transformation L R3 R3 such that 1 2 1 2 2

Find a linear transformation L: R3 -> R3 such that [ 1 -2 1 ], [ 2 2 4 ] is a basis for range L. please show all steps!

Find a basis for Ker L

Solution

Let L ( 1,0,0)^T = ( 1,-2,1)^T, L( 0,1,0)^T = (2,2,4)^T and L ( 0,0,1)^T = ( 1,-2,1)^T + (2,2,4)^T =          ( 3,0,8)^T Then L (v) = L( v₁ , v₂ , v₃ )^T = v₁ L ( 1,0,0)^T + v₂ L( 0,1,0)^T +v₃ L ( 0,0,1)^T = v₁ ( 1,-2,1)^T + v₂ (2,2,4)^T + v₃ ( 3,0,8)^T. The vectors ( 1,-2,1)^T and (2,2,4)^T are linearly independent and since ( 3,0,8)^T = ( 1,-2,1)^T + (2,2,4)^T, the vectors ( 1,-2,1)^T and (2,2,4)^T form a basis for range L. Ker L is the set of solutions of the equation L (v) = 0 i.e. v₁ ( 1,-2,1)^T + v₂ (2,2,4)^T + v₃ ( 3,0,8)^T = (0, 0, 0)^T The only solution of this equation is v₁ = 0, v₂ = 0 and v₃ = 0 i.e. a trivial solution. Since Ker L is the set of all vectors v such that L(v) = 0, we have Ker L = { (0,0,0)^T}.