Determine a basis of $\operatorname{Ker} \ F$

Determine a basis of $\operatorname{Ker} F$ and one for $\operatorname{Im} F$, where $F:\Bbb R^4\to \Bbb R^3$ is the linear transformation defined by $$F(x_1,x_2,x_3,x_4):=(x_1+x_2+x_3+x_4, 2x_2+x_3+x_4,4x_2+2x_3+2x_4) .$$

I have no idea how to start. Any idea please? Thank you.

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You could first find the matrix representation $A$ of $F$.

$A$ is the $3\times 4$ matrix whose $i^{\rm th}$ column is $F({\bf e}_i)$where ${\bf e}_i$ is the $i^{\rm th}$ unit vector in $\Bbb R^4$. You then have $$F({\bf x})=A{\bf x},$$ for all ${\bf x}\in \Bbb R^4$.

Then a basis for $\text{ ker}( F)$ is given by a basis for the null space of $A$ and a basis for the image of $F$ is given by a basis of the column space of $A$.

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Let's start with the kernel. The kernel is the set of inputs yielding the output zero. So for us that's the set of solutions to the system, \eqalign{x_1+x_2+x_3+x_4&=0\cr2x_2+x_3+x_4&=0\cr4x_2+2x_3+2x_4&=0\cr} Do you know how to find the solutions of such a homogeneous system of linear equations? Do you know how to find a basis for that set of solutions? If not, better learn it fast, as you'll be using that technique over and over and over in linear algebra.

By the way, there's nothing wrong with posting homework problems to this website, but, if this is a homework problem, you ought to add the "homework" tag.

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