Final answer:
To determine the standard matrix for the transformation w = A⋅x with given definitions for w₁, w₂, and w₃, you arrange the coefficients of x₁ and x₂ into a matrix. The resulting standard matrix is a 3x2 matrix with coefficients from the transformation equations as its entries.
Step-by-step explanation:
To find the standard matrix for the linear transformation defined by the equations w₁=-x₁+x₂, w₂=3x₁-2x₂, and w₃=5x₁-7x₂, you simply place the coefficients of x₁ and x₂ from each equation into a matrix, organizing them by rows.
The coefficients of x₁ form the first column of the matrix, and the coefficients of x₂ form the second column.
The standard matrix A for the transformation w = A⋅x is:
A =
\begin{pmatrix}
-1 & 1 \\
3 & -2 \\
5 & -7
\end{pmatrix}
This matrix can then be used to transform any vector x in the domain of the transformation to find the resulting vector w in the codomain.