Question

Consider the following linear transformation.\ $T(x, y) = (x + y, -x + y)$\ Find the standard matrix A for the linear transformation.\ A = \\ ??\ Find the inverse of A. (If an answer does not exist, enter DNE in any cell of the matrix.)\ A$^{-1}$ = \\ ??\ Determine whether the linear transformation is invertible. If it is, find its inverse. (If an answer does not exist, enter DNE.)\ $T^{-1}(x, y) = $ 

          Consider the following linear transformation.\
$T(x, y) = (x + y, -x + y)$\
Find the standard matrix A for the linear transformation.\
A = \\
??\
Find the inverse of A. (If an answer does not exist, enter DNE in any cell of the matrix.)\
A$^{-1}$ = \\
??\
Determine whether the linear transformation is invertible. If it is, find its inverse. (If an answer does not exist, enter DNE.)\
$T^{-1}(x, y) = $
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Consider the following linear transformation.T(x, y) = (x + y, -x + y)Find the standard matrix A for the linear transformation.A = ??Find the inverse of A. (If an answer does not exist, enter DNE in any cell of the matrix.)A^-1 = ??Determine whether the linear transformation is invertible. If it is, find its inverse. (If an answer does not exist, enter DNE.)T^-1(x, y) =

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Elementary and Intermediate Algebra
Elementary and Intermediate Algebra
Alan S. Tussy, R. David Gustafson 5th Edition
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Consider the following linear transformation T(x,y)=(x + y, x + y) Find the standard matrix A for the linear transformation. Find the inverse of A.If an answer does not exist,enter DNE in any cell of the matrix.
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Transcript

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00:01 In order to show that this matrix is invertible, let's find the kernel of this matrix.
00:06 Now the kernel of this matrix is given by those vectors, x1, x2, x3, such that x1 is equal to minus x3, x2 is equal to x3, and x1 minus x2 plus x3 is equal to 0.
00:24 Now, very easy algebraic computations are going to show that if a vector satisfies this, three conditions then it implies that x1 x2 x3 is just the zero vector so the kernel of this matrix here is zero and in particular since this one is a square matrix it means that it is invertible now let's find the matrix representing this linear map the matrix representing this linear map is 1 -0 -1 0 -2 oh sorry 0 1 minus 1 1 1 minus 1 1 now let's find the adjoined of this matrix okay the adjoint of this matrix is given by what okay here we are gonna have okay, one, this one multiplied by this one, minus this one multiplied by this one.
01:38 So is 0.
01:41 Then here what we're going to have, we are going to have this one multiplied by this one, which is 0, minus this one multiplied by this one, which is plus 1 here with minus sign.
01:57 So minus 1 then here what we are gonna have here we are gonna have this one multiply this one multiplied by this one minus this one multiplied by this one so minus 1 then here what we are gonna have here we're gonna have okay this one multiplied by this one minus this one so one then here we are gonna have okay one this one this one multiplied by this one minus this one multiplied by this one zero then here we are gonna have okay this one multiplied by this one which is minus one then here we're gonna have here we're gonna have minus one then here we're gonna have minus multiplied by minus 1, so 1.
03:05 And finally, here we're going to have 1...
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