The Chow groups CH^p(X,n) of an algebraic variety X were defined by Spencer Bloch in terms of certain codimension-p algebraic cycles on X x A^n; they are graded pieces of Quillen's higher algebraic K-groups K_n(X). In case X is a point, viewed as a variety over an infinite field F, there is a hope that the Chow groups of a field can be computed using only linear subvarieties, whereas the current definition uses essentially all algebraic cycles in the affine space. For example, this hope is true in the simplest non-trivial case CH^n(Spec F, n) as follows from the work of Burt Totaro, who showed that these groups are isomorphic to the Milnor K-groups K_n^M(F). In the talk we consider linear Chow groups of a field with low indices, discuss some related elementary geometric observations and prove an explicit relation between CH_linear^2(Spec F,3) and the Bloch group B(F).