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Kernal of a Transformation
Let $T: P_2\to \mathbb R_3$ be given by $T(a,b,c)=(a+b)t^2+(a+b+c)t+c$
I have already shown that this is Linear. I need to find the Kernal. I know this requires me to show when $T(a,b,c)$ is equal to the zero vector.
I need help doing this as we have only ever done this with matrix transformation and not these polynomial transformations.
The zero of $P_2(\mathbb{R})$ (polynomials of degree at most $2$ with real coefficients) is the zero polynomial. So, we must find all elements of $\mathbb{R}^3$ that map to the zero polynomial.
To do this, suppose we have an arbitrary element of $\mathbb{R}^3$ that maps to $0$, say $(a, b, c)$. Then we have
$$0\cdot t^2 + 0 \cdot t + 0 = T(a,b,c) = (a+b)t^2 + (a+b+c)t + c.$$
This gives us a system of 3 equations and 3 unknowns (feel free to edit my system in MathJax as I wasn’t sure how to do so). Namely,
$$0 = a+b, 0 = a+b+c, 0 = c.$$
A solution to this is $c=0, b = a, a \in \mathbb{R}$.
So, any vector that maps to th

The zero of $P_2(\mathbb{R})$ (polynomials of degree at most $2$ with real coefficients) is the zero polynomial. So, we must find all elements of $\mathbb{R}^3$ that map to the zero polynomial.
To do this, suppose we have an arbitrary element of $\mathbb{R}^3$ that maps to $0$, say $(a, b, c)$. Then we have
$$0\cdot t^2 + 0 \cdot t + 0 = T(a,b,c) = (a+b)t^2 + (a+b+c)t + c.$$
This gives us a system of 3 equations and 3 unknowns (feel free to edit my system in MathJax as I wasn’t sure how to do so). Namely,
$$0 = a+b, 0 = a+b+c, 0 = c.$$
A solution to this is $c=0, b = a, a \in \mathbb{R}$.
So, any vector that maps to the zero polynomial will have the form $(a, a, 0)$.
This tells us $$Ker(T) = \{(a, a, 0)  a \in \mathbb{R} \}.$$
20171214 07:44:14