$linear algebra$

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$linear algebra$

I need to find a non null R3 -> R3 linear transformation that transforms the vectors (1,1,1) ; (1,2,8) into null vectors. The question also asks for the dimension of the transformations image.

Linear Algebra

Kastro

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Take the map $f(x,y,z) = (6x-7y+z, 0, 0)$. It is not null as $f(1,0,0)=(6,0,0) \neq (0,0,0)$, but \[ f(1,1,1) = (6-7+1, 0, 0) = (0,0,0) \quad \text{and} \quad f(1,2,8) = (6-14+8, 0, 0) = (0,0,0). \] The dimension of the image is the dimension of the starting space (which is $3$) minus the dimension of the kernel (which is $\geq 2$ as it contains two independent vectors, but $\neq 3$ as the map is not null), so it's $1$.

Alessandro Iraci

$linear algebra$

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