How do you solve this linear transformation?
If we remember the matrix for a reflection, we just need a vector in the direction of the line. When x=3, y=-4 so use the vector 3i-4j = lx i + ly j = or (3,-4) what ever your notation is. The matrix is [lx^2-ly^2 2 lx ly] [2 lx ly ly^2-lx^2] and is normalized by 1/(lx^2+ly^2) I get T1:= [-7/25 -24/25] [-24/25 7/25] Now do the same for the matrix that reflects across -2x+3y=0 via the vector (3,2) T2:= [5/13 12/13] [12/13 -5/13] Now the matrix that implements the composition T1 o T2 is just T1*T2 I get T1 o T2 = [-323/325 36/325] [-36/325 -323/325] Now if you can't remember the formula, I would work out how any two different, non-zero points get moved (under the composition) and since everything is linear, you can solve for the matrix that accomplishes the transform. It's actually harder to work out both matrices and multiply. For instance, the point (325,325) gets mapped to (-287,-359). The point (650,325) gets mapped to (-610,-395) So T1oT2 [[325,650],[325,325]] = [[-287,-610],[-359,-395]] So right multiply both sides by the inverse [[325,650],[325,325]]^-1 = [[-1/325,2/325],[1/325,-1/325]] found using Kramer's rule. I get T1 o T2= [-323/325 36/325] [-36/325 -323/325] same as before. The rotation matrix has the form [cosθ sinθ] [-sinθ cosθ] for a counterclockwise rotation about the origin. cosθ=-323/325 and sinθ = 36/325 => θ = 3.0305956431... radians Use arccos and arcsin on your calculator and make sure the signs are right (i.e., you're in the right quadrant--or draw a picture).
Need help with a question about linear transformations?
A function is just a recipe to define a value on one variable to another variable. In this case both variables are defined on the same collection. A function is for instance: 1 -> 1 2 -> 2 3 -> 3 This is the identity function: it assigns to every value of the first variable (name = x) to the same value of the second variable (name this y) Every different assignment of values to the three original values is a function. For value 1 we can assign 3 values, for value 2 also 3 and for value 3 also 3. So there are 27 different functions. If you make an extra rule saying that every value of the result must reached in every function, a function like T: V -> {0} is not allowed. So than there are only 6 functions: You can assign 3 values to {0}, but than only 2 values are left for {1} and the value of {3} must be the value that is left over. So 3.2.1 possibilities: 6 functions. So the six functions are: {0} -> {0} and {1} -> {1} and {2} -> {2} {0} -> {0} and {1} -> {2} and {2} -> {1} {0} -> {1} and {1} -> {0} and {2} -> {2} {0} -> {1} and {1} -> {2} and {2} -> {0} {0} -> {2} and {1} -> {0} and {2} -> {1} {0} -> {2} and {1} -> {1} and {2} -> {0} All these 6 functions are 1-1 : every distinct value for x gives a distinct value for y. The inverse of each function is simple: reverse the arrows.
How do I show that L is a linear transformation?
Your idea of checking whether the relation [math]L[/math] satisfies the two axioms of a linear transformation was correct.For the first axiom:[math]L(f(x)+g(x)) = (f(x)+g(x))^\prime + (f(0)+g(0))[/math][math]= f^\prime(x) + g^\prime(x) + f(0) + g(0)[/math][math]= (f^\prime(x) + f(0)) + (g^\prime(x) + g(0))[/math][math]= L(f(x)) + L(g(x)).[/math]For the second axiom:[math]L(cf(x)) = (cf(x))^\prime + cf(0) = cf^\prime(x) + cf(0)[/math][math]= c(f^\prime(x)+f(0))[/math][math]= cL(f(x)).[/math]Hence, [math]L[/math] is a linear transformation.
I need help solving this linear transformation please.?
Hello Let T:R^2 →R^2 be a linear transformation that sends the vector u=(5,2) into (2,1) and maps v=(1,3) into (−1,3) . Use properties of a linear transformation to calculate T(-5u) = ( , ) T(8v) = ( , ) T(-5u+8v) = ( , )
Please help with linear transformation question!!!?
Suppose T: R^3 -> R^2 is a linear transformation such that T(1,0,0) = (4,5), T(0,1,0) = (-1,1), and T(2,1,-3) = (7,-1). Find the matrix of T. I am having a hard time figuring out how to convert the T(2,1,-3) into T(0,0,1). The answer in the back of the book says the matrix looks like this: 4 -1 0 5 1 4 So somehow T(0,0,1) equals (0,4) but I don't know how to get there. Please help me with this problem. Thank you in advance.
Linear transformation question?
I'm guessing this is a linear algebra question. To prove something is a linear transformation you must show it satisfies the following two conditions. A Linear Transformation T from the Real vector space V to the Real vector space W is a function from V to W which satisfies two conditions: 1. T(u+v) = T(u) + T(v) 2. T(k·v) = k·T(v), where "k" is a constant. It's an if and only if, thus you must prove both directions...assume b=0 and show T is a linear transformation. Then assume T is a linear transformation and show b must equal 0. Here is an example: For F(x)=2x we check the satisfability of the two conditions of a linear transformation: 1. F(x+y) = 2(x+y) = 2x + 2y = F(x) + F(y), "b/c we were given F(x)=2x...F(y)=2y" 2. F(k·x) = 2(k·x) = k·2x = k·F(x) Hence F(x) is a linear transformation from R to R. That should help. It's a bit late and don't want to go through the process of writing the proof to your problem. Plus...you will only learn it if you do the work. That's how math works.
Linear Transformation question please help!?
Find the rotation or reflection that is equal to the linear transformation which reflects every vector in R2 across the line y = x and then reflects across y-axis
How important are linear transformations in linear algebra?
Linear algebra is essentially the study of linear transformations. They are the most crucial, interesting, and useful part of linear algebra. Although linear transformations are between vector spaces, linear algebra isn’t called “vector algebra” for a reason.A vector space by itself isn’t very interesting — you’ve got vectors, you’ve got a dimension, you’ve got bases, you’ve got subspaces, that’s about it. Maybe you can play around with change-of-basis, but nothing particularly complicated or deep.A linear transformation is a map from one vector space to another, essentially a function on vectors that preserves linearity. Now you have lots of interesting questions: Can you compose linear transformations like functions? What is the range of a given linear transformation? What vectors does a linear transformation take to zero (the kernel of the linear transformation), and what properties does the kernel have? How can we tell if a linear transformation is invertible? How can we make it easier to compute the effects of a linear transformation? Is composition of linear transformations commutative? If not generally, what sort of transformations are? What are the invariants of a linear transformation? etc, etc, etc.Linear transformations allow you to write non-trivial equations on vectors and vector spaces, and linear algebra gives you techniques to solve them. Since linear transformations and vector spaces are an abstract subject, this can have broad applicability in fields outside linear algebra — differentiation is a linear operator on the vector space of functions, so differential equations can be solved using linear algebra techniques, for instance. Quantum mechanics models system states as vectors, and things like observation and stage transformations as linear operators, so linear algebra plays a big role in QM as well.