16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.1.2

Bilinear Forms

Mathematics — Linear Algebra · matrix-of-bilinear-form

Problem

Let ff be the bilinear form on R2R^2 defined by f((x1,y1),(x2,y2))=x1y1+x2y2.f((x_1, y_1), (x_2, y_2)) = x_1 y_1 + x_2 y_2. Find the matrix of ff in each of the following bases: {(1,0),(0,1)},{(1,−1),(1,1)},{(1,2),(3,4)}.\{(1, 0), (0, 1)\}, \quad \{(1, -1), (1, 1)\}, \quad \{(1, 2), (3, 4)\}.
Intuitive explanation

A bilinear form's matrix representation depends entirely on your choice of basis—the same form looks different in different coordinate systems. To find it, evaluate ff at each pair of basis vectors and arrange the results in a matrix: the (i,j)(i,j) entry is f(bi,bj)f(\mathbf{b}_i, \mathbf{b}_j). This is just a systematic way of recording how the form "sees" interactions between your basis elements.

Intuitive explanation

A bilinear form's matrix representation depends entirely on your choice of basis—the same form looks different in different coordinate systems. To find it, evaluate ff at each pair of basis vectors and arrange the results in a matrix: the (i,j)(i,j) entry is f(bi,bj)f(\mathbf{b}_i, \mathbf{b}_j). This is just a systematic way of recording how the form "sees" interactions between your basis elements.

Related theory

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