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Choose x = (x1, x2, x3, x4) in R4. It has 24 rearrangements like (x2, x1, x3, x4) and (x4, x3, x1, x2). Those 24 vectors, including x itself, span a subspace S. Find specific vectors x so that the dimension of S is: (a) 0, (b) 1, (c) 3, (d) 4.

Respuesta :

LammettHash

a. Take [tex]\mathbf x[/tex] to be the zero vector.

b. Fix one component [tex]x_i[/tex] from [tex]i\in\{1,2,3,4\}[/tex], then set [tex]x_j=0[/tex] for [tex]i\neq j[/tex]. One such choice would be [tex]\mathbf x=(0,x_2,0,0)[/tex].

c. Similar strategy as in (b), but we need 3 vectors in order for [tex]S[/tex] to have dimension of 3. One such choice would be [tex]\mathbf x\in\{(x_1,0,0,0),(0,x_2,0,0),(0,0,0,x_4)\}[/tex].

d. You can use the example from (c); then you just need to insert the vector [tex](0,0,x_3,0)[/tex].

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