<s>[INST] Image and kernel of a linear transformation I have a linear transformation:
$$f:\mathbb{R}^3 \rightarrow \mathbb{R}[x]_2$$
$$f((1,1,1))=2x^2-3x,\ f((1,2,3))=-3x,\ f((1,2,4))=2x^2-4x$$
I need to find kernel, image, dimension of image and dimension of kernel.
I tried to find formula describing this transformation:
$f((a,b,c))=(x_1a+y_2b+z_1c)x^2+(x_2a+y_2b+z_2c)x+x_3a+y_3b+z_3c$
$$\begin{cases} x_1+y_1+z_1=2 \\ x_1+2y_1+3z_1=0 \\x_1+2y_1+4z_1=2 \end{cases}$$
$$\begin{cases}x_2+y_2+z_2=-3 \\x_2+2y_2+3z_2=-3 \\x_2+2y_2+4z_2=-4 \end{cases}$$
$$\begin{cases}x_3+y_3+z_3=0 \\x_3+2y_3+3z_3=0 \\x_3+2y_3+4z_3=0\end{cases}$$
$$\begin{cases}x_1=6 \\ y_1=-6 \\ z_1=2 \\ x_2=-4 \\ y_2=2 \\ z_2=-1 \\ x_3=0 \\ y_3=0 \\ z_3=0 \end{cases}$$
$$f((a,b,c))=(6a-6b+2c)x^2+(-4a+2b-c)x$$
Kernel:
$$\begin{cases}6a-6b+2c=0 \\ -4a+2b-c=0\end{cases}$$
$$\begin{cases}b=-a \\ c=-6a\end{cases}$$
$$\ker(f)=\operatorname{Lin}((1,-1,6)) \\\dim\ker(f)=1$$
Unfortunately, I don't know how it is possible to find image (and its dimension) of $f$.
I think that $\dim\operatorname{Im}(f)=3-1=2$, but I am not sure about it.
 [/INST] Your expression for $f$ is correct. Here is a slightly easier way to get it:
$$f(0,0,1) = f(1,2,4) - f(1,2,3) = 2x^2 - x$$
$$f(0,1,0) = f(1,2,3) - f(1,1,1)-2\cdot f(0,0,1) = -6x^2 + 2x$$
$$f(0,0,1) = f(1,1,1) - f(0,1,0) - f(0,0,1) = 6x^2 - 4x$$
So
$$f(a,b,c) = a\cdot f(1,0,0) + b\cdot f(0,1,0) + c\cdot f(0,0,1) = (6a-6b+2c)x^2+(-4a+2b-c)x$$
Your calculations for the kernel are correct. $\{(1,-1,6)\}$ is the basis for $\operatorname{Ker} f$, and hence $\dim\operatorname{Ker} f = 1$.
Now, for the image, the set
$$\{f(1,0,0), f(0,1,0), f(0,0,1)\} = \{2x^2 - x, -6x^2 + 2x, 6x^2 - 4x\}$$
certainly spans $\operatorname{Im} f$. It only remains to be seen whether it is linearly independent. We obtain:
$$6x^2 - 4x = 6\cdot(2x^2-2) + (-6x^2 + 2x)$$
So
$$\operatorname{Im} f = \operatorname{span} \{2x^2 - x, -6x^2 + 2x, 6x^2 - 4x\} = \operatorname{span} \{2x^2 - x, -6x^2 + 2x\}$$
The set $\{2x^2 - x, -6x^2 + 2x\}$ is linearly independent so it is a basis for $\operatorname{Im}f$. We conclude that $\dim\operatorname{Im}f = 2$. 
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