Express a linear map in matrix form

Question

Is there anyway to express the linear map $x\mapsto (a^Tx)b, \, a,b,x\in\Bbb R^n$ in matrix form $x\mapsto Ax$?

Context: I'm trying to solve an equation in $x\in\Bbb R^n$: $$ab^Tx-(a^Tx)b=c$$ If I'm able to express $(a^Tx)b$ in martrix form $Ax$ I'm done (we don't need to care about existence or uniqueness of solutions since from other aspects it's guaranteed $(ab^T - A)$ will be invetible.

Best regards.

Answer 1

$a^Tx$ is a scalar, hence

$$(a^Tx) b= b(a^Tx)=(ba^T)x$$

Answer 2

Every linear map over finite dimensional vectors spaces can be written as a matrix. That's a rather fundamental concept in linear algebra, so it seems that you have learned the mechanics of linear algebra without really understanding the concepts of it.

Given a finite dimensional vector space V with basis vectors $\textbf{v}_1$, $\textbf{v}_2$, etc., every element v of V is defined by its coefficients $c_i$:

v = $c_1 \textbf{v}_1 + c_2 \textbf{v}_2 + ...$.

And given any linear transformation T: U ->V, T is uniquely defined by its action on the basis vectors of U. Thus, if for every i,j we know the ith coefficient of T($\textbf{u}_j$), then that defines T. A matrix is simply these coefficients arranged in an array. To find the matrix that represents a linear map with respect to some bases, you can simply apply T to each basis vector of U, take the coefficients of each result and arrange them in a column, and then arrange those columns in an array. That's what a matrix is.

In this case, the above isn't needed, because the transformation just scales the input (the matrix would thus be a scalar times the identity matrix), but in general, understanding what a matrix is makes working with linear algebra a lot easier.