Basis of Continuous Functions in R SinX
Get the answer to your homework problem.
Try Numerade free for 7 days
6. Consider V span{cos(r) , sin(x) } subspace of the vector space of continuous functions and linear transformation T : V _ V where T(f) = f(0) - cos(z) - f(3) - sin(r) . Find the matrix of T with respect to the basis {cos(z) + sin(x) , cos(z) sin(z)} and determine if T is an isomorphism: [6 points]
Related Question
6: Determine if the following subset of R3 is a subspace of R3 . Ifit is a subspace, then find a basis for the subspace: W = y 2x - Sz = 0}
Discussion
You must be signed in to discuss.
Video Transcript
in this question. We have to do if they're given subset of our cube is a subspace of r cubed. It is a subspace then we have to find that basis of the subway space. So that given Cetus W. Is equals to certo X. Why? And zed weird. It's satisfying the equation. two x 5. 0 should be the question zero. So this is given to us. Now If we take a set 000. So it will belongs to W. As on putting zero in it will get two multiplied by zero minus five, multiplied by zero which will be because of zero. So it is satisfying it. So thus this means that W is none empty. So it has elements in it. No we will let The Elements Margaret says X one vibrant 01 less X two By two. 02. Have we let these two mattresses which belongs to W. Thus we have that two X one -5. Z one should be close to zero and two X two minus five. You should also because zero they must satisfy the equation. So now we have matrix X one by 101. That's matrix X two by two. 02 Will be equal, Assume Matrix X one. Let's extra By one. That's why too 0- one. Place. 02. No it must satisfies the equation. So it will be too X one plus X two minus wife zed one plus zero. So it will be equals true. We can write it in the form to X one 501-plus 2. Extra -502. So Both these values are zero. So its value will come out to be zero. So from here we get that X one plus X two by one. So this mattress also belongs to them said W. Because it satisfies the equation. No, therefore we can write that W. S closed under edition. No we will let dr X by Z belongs to W. And CB any a scalar. So two X -50. It should be equal to zero. And ceo X by Z will be go shoes C. X. See by caesar. So it If we use the equation and you lose to see X -5 season. So it will be two times of two X -5. So the value of this is zero. So it will come out to be zero. So therefore ceo X. Y. Z also belongs to W. So therefore we can write that W. Is closed under scalar multi application. So it is closed under scalar multiplication also. So hence wh subspace of our cube. So it must be a service space of our Q. Because it is closed under scalar multiplication also. So now we will again let abc this mattress belongs to W. Be the arbitrary vector in W. Said so we have from air to a minus five CS echoes a zero which implies that too is who is he? Five C. So we will learn see North because 20 then go away It will be because of five multiplied by two. Okay so it will be questioned five. So hang lurk be because um Which is North Pecos zero. So from here we can write that matrix abc. Is he gonna do five K. M. And to. Okay so it will be costume care times off five zero to plus mm times of 01 and zero. We can write this matrix says this in this form. So w will be because you're span off 502 and zero one zero. So here we will let see one off 50 to plus. So you do of 010 Will be zero here. So from there we get the values of seven equals two C. Two as zero. So I see when in situ are coming to the to be zeros of the matrix 502 and 010 linearly independent. So they are linearly independent. So from here we see the basis of W. Will be 50 to Matrix and 010. My critics. So let us write that answer. Our answer to the question is for the first part. The answer is yes it is a subspace of that mattress. And we have to find the basis in the second parts of the bases are 502 and 01 zero. This is an required solution. Thank you sir, grimes
Source: https://www.numerade.com/ask/question/6-consider-v-spancosr-sinx-subspace-of-the-vector-space-of-continuous-functions-and-linear-transformation-t-v-_-v-where-tf-f0-cosz-f3-sinr-find-the-matrix-of-t-with-respect-to-the-basis-cosz-71283/
0 Response to "Basis of Continuous Functions in R SinX"
Post a Comment