Finding Inverse of linear transformation (basic linear algebra)

Question

Let $T$ be a linear transformation which is defined on $\mathbb{R}^3 $ by
$$T(x_1,x_2,x_3)=(3x_1,x_1-x_2,2x_1+x_2+x_3).$$ How can I find the inverse of $T$? How can i check that it even have an inverse? (Perhaps referring to some theorems that I'm not aware of) I'm confused.

Answer 1

You can solve this problem by using the definition of $T$ directly: You have the system of equations \begin{align}y_1 &= 3x_1 \\ y_2 &= x_1-x_2 \\ y_3 &= 2x_1+x_2+x_3.\end{align} To invert $T$, solve this system. If it doesn’t have a unique solution, then $T$ is not invertible.

If you prefer to work with matrices, then this is a matter of inverting the coefficient matrix $$A=\begin{bmatrix}3&0&0\\1&-1&0\\2&1&1\end{bmatrix}.$$ Observe that this is also the matrix of $T$ in the standard basis: if you expand $A(x_1,x_2,x_3)^T$ you will get exactly the three expressions that define $T$. Note also that you can tell immediately that $A$ is invertible: it is a triangular matrix, so its determinant is equal to the product of its main-diagonal entries. None of them are zero, so the product is nonzero.