Inverse Transformation Linear Algebra

Question

Find the inverse transformation if it exists. Your answers will contain y₁,y₂ and y₃

y₁ = x₁ + x₂

y₂ = -2x₁ + x₂ -2x₃

y₃ = 7x₁ - 3x₂ + 8x₃

x₁ =

x₂ =

x₃ =

Please help me solve this and show all steps with details so I can understand this.

Answer 1

First, you'd like to turn the system of equations into a coefficient matrix. This will look like:

[1, 1, 0]

[-2, 1, -2]

[7, -3, 8]

Now we can reconsider this question in terms of linear algebra: this problem can be reformulated y = Ax, where A is the coefficient matrix, y is the vector <y_1, y₂, y₃>, and x is the vector <x₁, x₂, x₃>.

The question would like you to solve for x₁, x₂, x₃ in terms of y_1, y₂, y₃. In order to do this, we want to find x = A^-1y. You should be able to compute A^-1y by means of row reduction on the augmented matrix [A | y], which means that you put the matrix A and the vector y together. The row reduction will yield [ I | A^-1 y], where I is the Identity matrix and A^-1y is the solution you're interested in.