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I don't quite understand the logic of this question here: If there is a linear transformation T, and T(f(t)) = f(3t-2), what is the det of T?

I first choose the basis 1, t, t^2, but I don't seem to be able to find the correlation between T(f) and f and how to go on from there.

So for example, given f(t)=1, how to find T(1) in relation to t?

user7544590
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3 Answers3

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Hint:

If we write $f(t) = a_0 + a_1 t + a_2 t^2$ then $T(f) = f(3t-2) = a_0 + a_1 (3t-2) + a_2 (3t-2)^2$ which can be written as

$$ T(f) = a_0 -2 a_1 + 4 a_2 + (3 a_1 - 12 a_2) t + 9 a_2 t^2 $$

This means we can write the transformation in terms of its effect on the coordinates of the polynomial as

$$ T : \begin{pmatrix} a_0 \\ a_1 \\ a_2 \end{pmatrix} \rightarrow \begin{pmatrix} a_0 -2 a_1 + 4 a_2 \\ 3 a_1 - 12 a_2 \\ 9 a_2 \end{pmatrix} $$ Can you work out a linear matrix and a determinant from here?

Paul Aljabar
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Starting with the first basis element $f(t):=1$, we find that $T({\bf 1})=1$. Next, for $f(t):=t$, we have $T({\bf t})=f(3t-2)=3t-2$. Similarly, $T({\bf t^2})=f(3t-2)=(3t-2)^2=9t^2-12t+4$.

Hence the columns of $[T]$ (matrix representation of $T$ wrt standard basis are $(1,\ 0, \ 0)$, $(-2, \ 3,\ 0)$ and $(4, \ -12, \ 9)$.

AnyAD
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  • I don't understand how you got T(1) = 1 just by assigning f(t) = 1, and also why f(3t-2) = 3t-2? – user7544590 Nov 07 '17 at 18:23
  • @user7544590 The transformation $T$ takes (as input) a function, say $g(t)$. The 'output' of this transformation is given to be the function $h(t):=g(3t-2)$. So basically the output of $T$ is another function (it is really the function $f$ scaled by $3$ and shifted right by $2/3$. The notation is a little confusing so you need to distinguish $1$ from the scaler $1$, since here it is a basis vector, the constant function that is equal to $1$ for all $t$. Scale and translate this function and you still get the same function. Similarly for the other basis vectors. – AnyAD Nov 08 '17 at 08:23
  • @user7544590 For $ f(t):={\bf t}$, we have $f(3t-2)=3t-2$. – AnyAD Nov 08 '17 at 08:26
  • what does the notation := mean? are you referring to the t in f(t) or assigning? because for f(3t-2) to equal 3t-2, f(t) needs to be equal to t right? And which vector are you referring to as the basis vector? – user7544590 Nov 10 '17 at 03:34
  • I use $f(t):=$ to say that I am denoting the function on the RHS by $f(t)$. I will do a simple example and post it as an answer. – AnyAD Nov 11 '17 at 03:52
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Suppose that $T$ is a linear transformation from the space of polynomials of degree less or equal $1$, into the space of polynomials of degree tell or equal to $2$ and that it is defined by [ T (p(x)=xp(x).] Then consider the matrix representations of $T$ with respect to the bases $\{{\bf1,x}\}$ and $\{{\bf 1,x,x^2}\}$. Then $T({\bf 1})=x$ and $T({\bf x})=x^2$. Then with the respect to given basis, $[T]$ is given by

\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}

AnyAD
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