MATB24H3 Lecture Notes - Linear Map, Row And Column Spaces, Symplectic Group
Document Summary
Let v and v be vector spaces over r. let t : v v be a linear transformation. Let b be a basis for v . Then for any v v , t (v) is uniquely determined by the vectors. To say that t (v) is uniquely determined by the vectors t (b) for all v v it is equivalent to saying that if t is a linear transformation and. T (b) = t (b) b b then, t = t. Since b is a basis of v . Then there are b1, b2, , bk b and r1, r2, , rk f such that v = r1b1 + r2b2 + + rkbk. T (v) = t (r1b1 + r2b2 + + rkbk) = r1t (b1) + r2t (b2) + + rkt (bk) = r1 t (b1) + r2 t (b2) + + rk t (bk) = t (r1b1 + r2b2 + + rkbk)
