The pre image of the point (-10,-6) under the transformation (x, y) →(x+3 y, y-x) is (a) (2,-4)

The pre image of the point (-10,-6) under the transformation...

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication. In problems involving transformations, each point in the domain is mapped to a point in the codomain based on linear combinations of its coordinates.

Inverse Transformation/Preimage

The inverse transformation, or preimage, is the process of determining the original input for a given output under a function. In the context of linear transformations, finding the preimage involves solving a system of linear equations that represent the transformation.

Systems of Linear Equations

Systems of linear equations arise naturally when solving for the preimage under a linear transformation. Each component of the transformed output gives rise to an equation, and solving this system yields the original coordinates from which the output was obtained.

Transcript

00:01 So in the given question we are told that the pre -image of a point, the three image of the point minus 10 minus 6 under the transformation, under the transformation which is given as the transformation is given as x, from xy it would change to x plus 3y x plus 3y and y minus x so under this transformation what is the pre image of this point so this is what is asked in the question so let's assume that the pre image of the point is xy and after transformation we have the point x dash x dash y dash right so we can use the matrix in order to transform this point to x dash y dash since we already know the transformation right so what we can write is we can write the transformation as in the matrix form as 1 3 minus 1 1 times x y would give us the transformed point which is x dash dash and now what we can do is we can take we need to find x y right we already know x dash y dash which is minus 10 minus 6 so to find the pre image of x x dash y dash what we would need is x dash the inverse of we can take this matrix to the other side and and we would have the inverse of this matrix times x dash y dash is the required matrix xy.

02:16 So to find the inverse of a 2 by 2 matrix which is given us a, b, c, d, what we can do is we can take 1 by ad minus b c times the matrix d minus c.

02:37 So using this formula we can write the inverse of 1 3 minus 1 1 is equal to 1 by ad which is 1 times 1 minus b c which is minus minus 1 times 3 which is minus 3 and minus of minus 3 is plus 3 and in the matrix we would have 1 minus 3 minus minus minus minus of minus 1 which is 1 and a which is 1 so this is what we would have which is equal to 1 by 4 times 1 minus 3 1 1 right so now what we can do is we can substitute for x dash y dash as minus 10 minus 6 and and then we would have two matrices which are given like this...

Please give Ace some feedback