Questions and Answers #6 Matrix Transformations

Questions and Answers Sheet 6 Matrix Transformations Question #1 Given Matrix transformation matrix 111 (4-022) 003 Are there vectors in R that do not change under this transformation, ie, such that T(u)=u? Explain why yes or why not Answer: Step 1 Let (21-0 0 for any A A little work shows that these are the only solutions to Au=u A note: no matter what linear transformation you have, T(0)=0 is always true, but sometimes students forget to state the whole problem, so I posted a complete set of solutions just in case Question #2 From my working, P would equate to 0 P- 0 Now to find the transformation shown by P, would I have to cube the matrix or is there something that I'm missing here? Answer: Step 1 You can compute P or use the fact that if P rotates vectors through an angle ff, then P rotates vectors through the angle in 6 Question #3 Lets say we have the following matrix (M Then I want to transform all numbers in the matrix with the following function: f(x)=(x-4)²+2x-4 What would be the correct notation for this? Answer: Step 1 Could use the "Hadamard Product". I don't know whether there is a consistently accepted notation for such a product, however a few examples here and on Wikipedia seem to use an open circle. In that sense, your expression would take the form (M-40) (M-40)+2M-40 where (0) In MATLAB, use. for element-wise multiplication. There are probably other possibilities. Question #4 Find the matrix representation of the linear transformation T that maps the input vector output vector (2.-3) and maps the input vector (1,2) to the output vector (1,1) to the (5.1) Answer: Step 1 Suppose the matrix is V, then we have Consider to find the inverse matrix of Question #5 Let (e-()) and ( 0 1 and (V=()) 8 Let T:RR be a linear transformation that maps e to y, and e to y Find the image of (( Answer: Step 1 Yes, your thought on how to approach it is correct. Every linear transformation is uniquely determined by its action on a basis. In this case, suppose the matrix of the transformation is So you know bl Does that tell you anything about the matrix A Question #6 Finding an expression for x and y variables in terms of the corresponding image X and Y variables, find the image of the line y=2x-7 under the transformation given by (M=( 32 Answer: So, if y=2x-7, then X=x-3y=-5x+21 and Y=-3x+2y=x-14 Question #7 Find the Image 2x-3y=0 given matrix transformation 52 4 (T- Answer: The line 2x-3y=0 defines a subspace. The vector r that you used is orthogonal to this line. The correct point of view is to look at the line as set of all (x,y) € R² such that (x,y) = (xx) since y=x 523 412 So it suffices to check where T sends, say (3.2) (which is on the line). Thus, computing should give the answer. Question #8 Do elementary row operations give a similar matrix transformation? So we define two matrices A, B to be similar if there exists an invertible square matrix P such that AP=PB Answer: No. For instance, because row permutations do not preserve the trace. 2 3 2 you can charige the first matrix to the second via a permutation of the rows, but the trace of the first one is 4 and the trace of the second one is 6, If AP=PB then trBtr(PAP)tr(PPAP) = tr Question #9 How to find the linear transformation associated with a given matrix? It is well known that given two bases (or even one if we consider the canonical basis) of a vector space. every linear transformation 7:VW can be represented as a matrix, but since this is an isomorphism between L(V.W) and Mx where the latter represents the space mitimes n matrices on the same field in which are defined respectively vector spaces. Answer: I am not sure what you expect, say T is your linear transform, and A represents it in the basis for V= [aa] and W Then we say Ta = ΣΑ (b.bl This should completely define what T "does". Typically, one defines the action of T on the basis vectors al in order to determine what T "looks like". Otherwise, one can not write down much about T. Please clarify what type of answer you are expecting if this is not sufficient. Question #10 I have a 3d object, to which I sequentially apply 3 4 x 4 transformation matrices, A, B, and C. To generalize, each transformation matrix is determined by the multiplication of a rotation matrix by a translation matrix. How can I calculate the final transformation matrix 1, which defines how to get from the original 3d object to the final transformed object? Answer: If you take an initial point pi, apply a transform A, then apply transform B, then apply transform C to get a final point py, then we have Pr = C(B(A(p))) PCBAP So the total transformation is t=CBA This is one of the reasons why matrix multiplication is so nice-it lets us compose many linear transforms into a single linear transform. You just need to remember that the first transformation belongs on the right (closest to the point p it acts on).