In the vector space consisting of real polynomials of a degree no more than 2, the following subset is considered: determine a linear transformation that has as a kernel. show and explain what you do along the way, step-by-step : -)if you write the answer in hand, will you be kind to write as clearly as possible. i would be very grateful for that and will appreciate the you can give : -)

We could express any vector as

So any vector would have coefficients that satisfy

from which we can show that any such vector is a linear combination of some other vectors:

More explicitly, we've shown that is a linear combination of the vectors and ; in other words, , and in fact these two vectors form a basis for . But this set does not span all of because there's no combination of these two vectors that can be used to obtain a constant. We want the transformation to be usable for any vector in , so we need to add an additional vector extend the basis. We can do this simply by appending into the spanning set. (Do check that the vectors remain linearly independent.)

Now we want the transformation to map those polynomials for which to the zero vector. We know which vectors belong to the basis of , so we need

where the choice of the assignment for is arbitrary, so long as it's non-zero.

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step-by-step explanation: