Page 323 - 35Linear Algebra
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                      8. What are the four main things we need to define for a vector space? Which
                         of the following is a vector space over R? For those that are not vector
                         spaces, modify one part of the definition to make it into a vector space.

                          (a) V = { 2 × 2 matrices with entries in R}, usual matrix addition, and

                                   a b          ka b
                              k ·          =            for k ∈ R.
                                   c d          kc d
                          (b) V = {polynomials with complex coefficients of degree ≤ 3}, with usual
                              addition and scalar multiplication of polynomials.
                                               3
                          (c) V = {vectors in R with at least one entry containing a 1}, with usual
                              addition and scalar multiplication.

                      9. Subspaces: If V is a vector space, we say that U is a subspace of V when the
                         set U is also a vector space, using the vector addition and scalar multiplica-
                         tion rules of the vector space V . (Remember that U ⊂ V says that “U is a
                         subset of V ”, i.e., all elements of U are also elements of V . The symbol ∀
                         means “for all” and ∈ means “is an element of”.)

                         Explain why additive closure (u + w ∈ U ∀ u, v ∈ U) and multiplicative
                         closure (r.u ∈ U ∀ r ∈ R, u ∈ V ) ensure that (i) the zero vector 0 ∈ U and
                         (ii) every u ∈ U has an additive inverse.



                         In fact it suffices to check closure under addition and scalar multiplication
                         to verify that U is a vector space. Check whether the following choices of U
                         are vector spaces:

                                                
                                   x              
                                      y
                          (a) U =       : x, y ∈ R
                                      0
                                                  
                                                
                                   
                                   1           
                          (b) U =       : z ∈ R
                                      0
                                      z
                                               
                     10. Find an LU decomposition for the matrix

                                                                    
                                                      1   1 −1      2
                                                      1   3    2    2
                                                                    
                                                                    
                                                    −1 −3 −4        6
                                                                    
                                                      0   4    7 −2
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