Determinants

A linear transformation from R2 to R is a real function F of two real variables such that (1.1)<math display='block'> <mrow> <mi>F</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>+</mo><mi>x</mi><mo>&#x0027;</mo><mo>,</mo><mi>y</mi><mo>+</mo><mi>y</mi><mo>&#x0027;</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>+</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>&#x0027;</mo><mo>+</mo><mi>y</mi><mo>&#x0027;</mo><mo stretchy='false'>)</mo><mo>,</mo><mtext>&#x2003;</mtext><mi>F</mi><mo stretchy='false'>(</mo><mi>t</mi><mi>x</mi><mo>,</mo><mi>t</mi><mi>y</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>t</mi><mi>F</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo> </mrow> </math>$$F(x + x',y + y') = F(x,y) + F(x' + y'),\quad F(tx,ty) = tF(x,y)$$ for all real numbers x, y, x′, y′, t. The only such functions are (1.2)<math display='block'> <mrow> <mi>F</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi> </mrow> </math>$$F(x,y) = ax + by$$ where a, b are real constants.