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"For linear transformations $ \colon : P_2 \rightarrow P_2$ ($P_2$ is the space of polynomials of degree $2$ or less) find a matrix."

a) $S(f)(x) = f(2x+2)$

b) $S(f)(x) = f(2)x$

Can somebody help me understand what this professor is expecting? Nobody in the math center at my university could tell me how it is done. Thanks

levap
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user2905256
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  • This is quite badly worded problem, but I would guess the meaning is "Find the matrix representing the linear map $S$ with respect to a certain basis $\beta$ of $P_2(\mathbb{R})$". Since $\beta$ is not specified, I would guess you should work with the "most standard" basis $\beta = (1,x,x^2)$ and get two $3 \times 3$ matrices which represent each of the maps with respect to $\beta$. – levap Mar 01 '17 at 01:47
  • The map for (a) seems to be $ax^2 + bx + c \mapsto a(2x+2)^2 + b(2x+2) + c$. While the map for (b) is $ax^2 + bx+c \mapsto (a(2)^2 + b(2)+c)x$. – Bobbie D Mar 01 '17 at 01:49

2 Answers2

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The formulas in parts (a) and (b) define linear transformations on the vector space $P_2$ of polynomials of degree at most 2. The question is asking you to write down a matrix representation of these linear transformations. However, the question is incomplete as stated: to write down the matrix that represents a linear transformation, you need to know bases of both the domain and codomain of the linear transformation.

In this case, your instructor probably intends for you to use the standard basis $\left\{1, x, x^2\right\}$ (for both the domain and the codomain, which in this case are identical).

Now, to determine the matrix that represents $S$ in part (a), for example, all you need to do is determine how $S$ acts on the basis elements. Note that $S(1)=1$, $S(x)=2x+2$, and $S(x^2)=(2x+2)^2=4x^2+8x+4$. You ought to be able to use this information to write down the matrix representation of $S$ with respect to the standard basis.

symplectomorphic
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  • Now when I find the matrix that S represents, that's the answer, correct? Sorry, my instructor is very vague with everything he teaches – user2905256 Mar 01 '17 at 01:53
  • @user2905256: yes, the answer to each question, part (a) and part (b), is a matrix. (They will be different matrices.) – symplectomorphic Mar 01 '17 at 01:55
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Let's do a), as an example: we want to find the matrix of $S(f)(x) = f(2x+2)$ with respect to the basis $\mathcal B = \{p_0(x),p_1(x),p_2(x)\}$ with $p_0(x) = 1,p_1(x) = x,p_2(x) = x^2$ (we could choose a different basis, but this is the simplest choice). We find that $$ S[p_0](x) = p_0(2x + 2) = 1 = 1\cdot p_0(x) + 0 \cdot p_1(x) + 0 \cdot p_2(x) $$ So, we find that the first column of our matrix is $(1,0,0)$. Similarly, $$ S[p_0](x) = p_2(2x+2) = (2x+2)^2 = 4 + 8x + 4x^2 = 4\cdot p_0(x) + 8 \cdot p_1(x) + 4 \cdot p_2(x) $$ So, we find that the third column of our matrix is $(4,4,1)$. Our final answer to this question will look something like this: $$ [S]_{\mathcal B} = \pmatrix{1&?&4\\0&?&8\\0&?&4} $$ I will leave it to you to find the missing column. Note that we might have gotten a different matrix if we had chosen a different basis, but this other matrix would still be valid.

Ben Grossmann
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  • Could you please verify for me that the missing column is (2,2,0)? – user2905256 Mar 01 '17 at 13:47
  • Yes, that's right – Ben Grossmann Mar 01 '17 at 13:48
  • Thank you. Do I need to put it in REF or is that matrix fine as the answer? Thanks again – user2905256 Mar 01 '17 at 13:50
  • What would you accomplish by putting this matrix in REF? – Ben Grossmann Mar 01 '17 at 13:51
  • Not sure. That's most of my frustration with this class. Nobody understands what the professor is saying so we are all just trying to blindly follow the steps. We don't have any reasons for doing anything, which is a terrible way to learn math in my opinion. I do appreciate your help though – user2905256 Mar 01 '17 at 13:53
  • @user2905256 No, the matrix as we've found it is the answer. Sometimes when a student complains about a professor being vague/unclear, he's right. The majority of the time, however, it is the student who is at fault, so forgive me if I take your opinions with a good deal of skepticism. – Ben Grossmann Mar 01 '17 at 13:56
  • @user2905256 Linear algebra (and the particular material your professor has chosen) is tricky. Many students are used to asking/learning how to get from problem statement to solution and so will ask "what are the steps? You've said a bunch of stuff, but what do I do?". That's fine for the math classes leading up to linear algebra, where everything is an algebra/differentiation/integration problem. With linear algebra, however, one needs to really consider why certain steps are taken. – Ben Grossmann Mar 01 '17 at 14:02
  • @user2905256 The fact that you might consider taking the REF of this matrix makes it clear to me that you have a good idea of how to row reduce (and that you had a few chapters demanding that you row-reduce over and over again), but that you don't really understand what row-reduction does. That is, you don't really understand why one should row-reduce in the first place. It could be that the professor has neglected to explain this, but it's far more likely that you and your classmates considered the "why" irrelevant, and so never learned it. – Ben Grossmann Mar 01 '17 at 14:07
  • I actually find the why very interesting and it seems like it would be an invaluable tool for solving systems and coming up with your own. I actually was excited for this class so that I could apply some of the knowledge to game development. If you have any resources that you know are particularly helpful, I would love to know of them. – user2905256 Mar 01 '17 at 14:13
  • @user2905256 what textbook does your class use? – Ben Grossmann Mar 01 '17 at 14:15
  • Linear Algebra- a Modern Introducton, David Poole. The book uses a lot of the same notation as my professor, (Probably why he likes it) but I think it would make more sense to me if could see how it actually related to the equations and graphs. If I had some sort of application. At least the book has examples, the prof. just does proofs. – user2905256 Mar 01 '17 at 14:18
  • @user2905256 first of all, I highly recommend this youtube series not as a study guide, but as a look at the big picture; should be about 2.5 hours long. Linear algebra is about so much more than solving systems of equations, and doesn't have that much to do with "graphs" in the calculus/algebra sense – Ben Grossmann Mar 01 '17 at 14:23
  • @user2905256 this page has a lot of recommendations. It's pretty low on the list, but I'd say that Lay's "Linear Algebra and its Applications" is probably the most readable of the bunch. If you're looking for some supplementary resources, Lay will probably be the most helpful; check a university library. I learned out of Friedberg-Insel-Spence, which can be challenging, and I've taught out of Gilbert Strang, which can be unclear if you don't "catch on". – Ben Grossmann Mar 01 '17 at 14:29
  • @user2905256 And, Strang can be wordy. These notes have plenty of worked examples, but they don't cover the "matrix of a linear transformation" stuff that your professor is covering. My bet is that this will make the eigenvalue/eigenvector stuff look very different. – Ben Grossmann Mar 01 '17 at 14:36
  • Thank you so much! I'll make sure to look over those when I get time. Can't tell you how great it is to have an someone care enough to help, when my instructor won't even show up to office hours. You are what a professor should be! – user2905256 Mar 01 '17 at 14:42