Linear transformationsThroughout,V and W are vector spaces.Definition 1. A map T : V → W is a linear transformation ifT (cx + dy) = cT (x) + dT (y)for all x, y in V and all c, d in R.Example 2. Let A be an m × n matrix.Then the mapT (x) = Ax is a linear transforma tion T : Rn→ Rm.Why?Example 3. Let Pnbe the vector space of all polynomials of degree at most n. Considerthe map T : Pn→ Pn− 1given byT (p(t)) =ddtp(t).This map is linear! Why?Representing linear maps by matricesLet x1,, xnbe a basis for V .A linear mapT : V → W is determined by the values T (x1),, T (xn).Why?Armin [email protected] 4. (From linear maps to matrices)Let x1,, xnbe a basis for V , and y1,, yma basis for W .The matrix representingT with respect to the se bases• has n columns (one for each of the xj),• the j-th column has m entries a1,j,, am, jdetermined byT (xj) = a1, jy1++ am,jym.Example 5. Let V = R2and W = R3. Let T be the linear map such thatT10=123, T01=407.What is the matrix A(T ) representing T with respect to the standard bases?Solution.Armin [email protected] 6. As in the previous example, let V = R2and W = R3. Let T be the linearmap such thatT10=123, T01=407.What is the matrix B(T ) representin g T with respect to the following bases?11x1,−12x2for R2,111y1,010y2,001y3for R3.Solution.Armin [email protected] matrix representing T en codes in column j the coefficients of T (xj) expressed asa linear combination of y1,, ym.Example 7. Let T : P3→ P2be the linear map given byT (p(t)) =ddtp(t).What is the matrix A(T ) representing T with respect to the standard bases?Solution.Armin [email protected] geometric examplesWe consider some linear mapsR2→ R2and their geometric interpretation.Example 8. The matr ix A =c 00 cExample 9. The matr ix A =0 11 0Example 10. The matrix A =1 00 0Example 11. The matrix A =0 −11 0Armin [email protected] 12. The matrix A =cosθ −sinθsinθ cosθExample 13. Let T be the li near map which projects each vect or onto the line withslope θ.• Which matrix represents T (with respect to the stand ard basis)?• Give a basis of R2with respect to which T is represented by a very simple matrix.Solution.Armin
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