Page 328 - 35Linear Algebra
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328                                                                       Sample First Midterm



                                             I kI
                                       k
                                  so, M =           , or explicitly
                                             0   I
                                                                               
                                                                1 0 0 k 0 0
                                                                0 1 0 0 k 0
                                                                               
                                                                               
                                                                0 0 1 0 0 k     
                                                          k
                                                              
                                                       M =                       .
                                                               0 0 0 1 0 0     
                                                                               
                                                               0 0 0 0 1 0     
                                                                0 0 0 0 0 1
                               7. We can call a function f : V −→ W linear if the sets V and W are vector
                                  spaces and f obeys
                                                       f(αu + βv) = αf(u) + βf(v) ,

                                  for all u, v ∈ V and α, β ∈ R.
                                  Now, integration is a linear transformation from the space V of all inte-
                                  grable functions (don’t be confused between the definition of a linear func-
                                  tion above, and integrable functions f(x) which here are the vectors in V )
                                                                 R  ∞                       R  ∞
                                  to the real numbers R, because     (αf(x) + βg(x))dx = α       f(x)dx +
                                   R  ∞                           −∞                         −∞
                                  β     g(x)dx.
                                    −∞
                               8. The four main ingredients are (i) a set V of vectors, (ii) a number field K
                                  (usually K = R), (iii) a rule for adding vectors (vector addition) and (iv)
                                  a way to multiply vectors by a number to produce a new vector (scalar
                                  multiplication). There are, of course, ten rules that these four ingredients
                                  must obey.

                                   (a) This is not a vector space. Notice that distributivity of scalar multi-
                                       plication requires 2u = (1 + 1)u = u + u for any vector u but


                                                                  a  b      2a  b
                                                             2 ·        =
                                                                  c d       2c d
                                       which does not equal


                                                          a  b      a  b      2a   2b
                                                                +          =           .
                                                          c d       c d        2c 2d
                                       This could be repaired by taking


                                                                a   b      ka  kb
                                                            k ·        =            .
                                                                c d        kc kd
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