Linear Algebra

158 Chapter Three. Maps Between Spacesthen this correspondence preserves the operations, for instance this addition( ) ( ) ( )1 3 4(1 2) + (3 4) = (4 6) ←→ + =2 4 6and this scalar multiplication.( ) ( )1 55 · (1 2) = (5 10) ←→ 5 · =2 10More generally stated, under the correspondence( )a 0(a 0 a 1 ) ←→a 1both operations are preserved:(a 0 a 1 ) + (b 0 b 1 ) = (a 0 + b 0 a 1 + b 1 ) ←→( ) ( ) ( )a 0 b 0 a 0 + b 0+ =a 1 b 1 a 1 + b 1andr · (a 0 a 1 ) = (ra 0 ra 1 ) ←→ r ·( ) ( )a 0 ra 0=a 1 ra 1(all of the variables are real numbers).1.2 Example Another two spaces we can think of as “the same” are P 2 , the spaceof quadratic polynomials, and R 3 . A natural correspondence is this.⎛ ⎞⎛ ⎞a 01a 0 + a 1 x + a 2 x 2 ⎜ ⎟←→ ⎝a 1 ⎠ (e.g., 1 + 2x + 3x 2 ⎜ ⎟←→ ⎝2⎠)a 2 3This preserves structure: corresponding elements add in a corresponding way⎛a 0 + a 1 x + a 2 x 2+ b 0 + b 1 x + b 2 x 2 ⎜a ⎞ ⎛0⎟ ⎜b ⎞ ⎛ ⎞0 a 0 + b 0⎟ ⎜ ⎟←→ ⎝a 1 ⎠ + ⎝b 1 ⎠ = ⎝a 1 + b 1 ⎠(a 0 + b 0 ) + (a 1 + b 1 )x + (a 2 + b 2 )x 2 a 2 b 2 a 2 + b 2and scalar multiplication also corresponds.⎛r · (a 0 + a 1 x + a 2 x 2 ) = (ra 0 ) + (ra 1 )x + (ra 2 )x 2 ⎜a ⎞ ⎛0⎟ ⎜ra ⎞0⎟←→ r · ⎝a 1 ⎠ = ⎝ra 1 ⎠a 2 ra 21.3 Definition An isomorphism between two vector spaces V and W is a mapf: V → W that(1) is a correspondence: f is one-to-one and onto; ∗