Linear system as unknown combination of columns

The system $A\mathbf{x}=\mathbf{b}$ asks which linear combination of columns of $A$ produces $\mathbf{b}$. Write $A=[\mathbf{a}_1\ \cdots\ \mathbf{a}_n]$; then $\mathbf{b}=x_1\mathbf{a}_1+\cdots+x_n\mathbf{a}_n$ . Geometric Cramer ratios compare signed volumes rather than manipulating rows blindly.

Cramer's rule requires invertible square $A$ (equivalently $\det(A)\neq0$). Each coordinate is a ratio of two determinants: replace one column by $\mathbf{b}$ in the numerator, divide by $\det(A)$.

Column picture reminder: if $\mathbf{a}_1,\mathbf{a}_2$ span a parallelogram in the plane, solving $x_1\mathbf{a}_1+x_2\mathbf{a}_2=\mathbf{b}$ asks how much of each column direction is needed to reach $\mathbf{b}$ .

This chapter is cultural insight, not a stable numerical algorithm. Practical large linear solves use LU or QR factorizations with pivoting, not cofactor expansions that grow factorially in cost.

Numeric analysts avoid Cramer for large $n$ because cofactor expansion costs scale like $n!$ and numerical stability is worse than Gaussian elimination with partial pivoting.

Treat Cramer as a microscope for small systems: it shows how each coordinate compares two signed volumes, not as a production solver for large matrices.