Linear Algebra

Section III. Computing Linear Maps 207̌ 2.19 Because the rank of a matrix equals the rank of any map it represents, ifone matrix represents two different maps H = Rep B,D (h) = RepˆB, ˆD(ĥ) (whereh, ĥ: V → W) then the dimension of the range space of h equals the dimension ofthe range space of ĥ. Must these equal-dimensioned range spaces actually be thesame?̌ 2.20 Let V be an n-dimensional space with bases B and D. Consider a map thatsends, for ⃗v ∈ V, the column vector representing ⃗v with respect to B to the columnvector representing ⃗v with respect to D. Show that map is a linear transformationof R n .2.21 Example 2.3 shows that changing the pair of bases can change the map thata matrix represents, even though the domain and codomain remain the same.Could the map ever not change? Is there a matrix H, vector spaces V and W,and associated pairs of bases B 1 , D 1 and B 2 , D 2 (with B 1 ≠ B 2 or D 1 ≠ D 2 orboth) such that the map represented by H with respect to B 1 , D 1 equals the maprepresented by H with respect to B 2 , D 2 ?̌ 2.22 A square matrix is a diagonal matrix if it is all zeroes except possibly for theentries on its upper-left to lower-right diagonal — its 1, 1 entry, its 2, 2 entry, etc.Show that a linear map is an isomorphism if there are bases such that, with respectto those bases, the map is represented by a diagonal matrix with no zeroes on thediagonal.2.23 Describe geometrically the action on R 2 of the map represented with respectto the standard bases E 2 , E 2 by this matrix.( ) 3 00 2Do the same for these.( 1) 0( 0) 1( 1) 30 0 1 0 0 12.24 The fact that for any linear map the rank plus the nullity equals the dimensionof the domain shows that a necessary condition for the existence of a homomorphismbetween two spaces, onto the second space, is that there be no gain in dimension.That is, where h: V → W is onto, the dimension of W must be less than or equalto the dimension of V.(a) Show that this (strong) converse holds: no gain in dimension implies thatthere is a homomorphism and, further, any matrix with the correct size andcorrect rank represents such a map.(b) Are there bases for R 3 such that this matrix⎛ ⎞1 0 0H = ⎝2 0 0⎠0 1 0represents a map from R 3 to R 3 whose range is the xy plane subspace of R 3 ?2.25 Let V be an n-dimensional space and suppose that ⃗x ∈ R n . Fix a basisB for V and consider the map h ⃗x : V → R given ⃗v ↦→ ⃗x • Rep B (⃗v) by the dotproduct.(a) Show that this map is linear.(b) Show that for any linear map g: V → R there is an ⃗x ∈ R n such that g = h ⃗x .(c) In the prior item we fixed the basis and varied the ⃗x to get all possible linearmaps. Can we get all possible linear maps by fixing an ⃗x and varying the basis?