Define the linear transformation T: R^n -> R^m by T(v) = Av. Use the matrix A to a) determine th

Name: Define the linear transformation T: R^n -> R^m by T(v) = Av. Use the matrix A to a) determine the dimensions of R^n and R^m, (b) find the image of v, and (c) find the pre-image of w. A = [[1, 2, -1], [1, 0, 1]], v = (5,2,2), w = (4,2)
Uploaded: 2022-10-08T14:35:58-08:00
Duration: 5 min 27 s
Channel: Adi S
Description: Define the linear transformation T: R^n -> R^m by T(v) = Av. Use the matrix A to a) determine the dimensions of R^n and R^m, (b) find the image of v, and (c) find the pre-image of w. A = [[1, 2, -1], [1, 0, 1]], v = (5,2,2), w = (4,2)

Step 1: Determine the dimensions of R^n and R^m.

Define the linear transformation T: R^n -> R^m by T(v) = Av....

Define the linear transformation T: R^n -> R^m by T(v) = Av. Use the matrix A to a) determine the dimensions of R^n and R^m, (b) find the image of v, and (c) find the pre-image of w. A = [[1, 2, -1], [1, 0, 1]], v = (5,2,2), w = (4,2)

Transcript

00:01 In this problem we are given that t is a linear transformation from the vector space rn to the vector space rm defined as t of v is equal to av where a is the matrix 1 to negative 1, 101.

00:21 The first question is to determine the dimensions of rn and rm.

00:30 Now if t is a linear transformation from a vector space rn to vector space rm and if a is the matrix corresponding to the linear transformation t, then a will be an m by n matrix where m is the dimension of rm, denoted as dimension of rm and n is the dimension of rn.

00:58 So here it is given that a is the metric corresponding to the linear transformation t from rn to rm and see that a is a 2 by 3 matrix and this means that dimension of rm is 2 and dimension of rn is 3.

01:29 Since a is a 2 by 3 matrix.

01:34 So these are the required first answers.

01:39 The second question is to determine the image under the linear transformation t of the vector v, which is the vector 522.

01:51 So we know that t of v is equal to av.

01:55 So that t of 522 will be the vector a which is the vector 1 to negative 1 1 1 .101 times the vector v which is the vector 522.

02:10 So perform this matrix multiplication.

02:14 We have this is the matrix 5 times 1 plus 2 times 2 plus negative 1 times 2 and the second row is 5.

02:25 5 times 1 plus 0 times 2 plus 1 times 2.

02:30 Therefore we get the image of the vector v is the vector 7 -7.

02:39 Therefore this is the required vector that will be the image vector...

Question

Define the linear transformation T: R^n -> R^m by T(v) = Av. Use the matrix A to a) determine the dimensions of R^n and R^m, (b) find the image of v, and (c) find the pre-image of w. A = [[1, 2, -1], [1, 0, 1]], v = (5,2,2), w = (4,2)

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