The symmetric product
Where the wedge demands sign flips, the symmetric algebra demands the opposite: v . w = w . v. Quotient the tensor algebra by the relations v tensor w - w tensor v = 0 and the graded pieces Sym^k V become the space of homogeneous degree-k polynomials in the basis vectors. Sym^2 V is exactly the home of quadratic forms — the things Vol I wrote as u^T B v with symmetric B.
For n = dim V = 3:
dim Sym^k V = C(n+k-1, k).
Sym^2 of R^3 has dim C(4,2) = 6, basis:
x^2, y^2, z^2, xy, xz, yz (the symmetric monomials)
-> exactly the 6 free entries of a symmetric 3x3 matrix.
Compare the two quotients of degree 2 over R^3:
dim Sym^2 V = 6 (symmetric part: T_{ij} = T_{ji})
dim Lambda^2 V = 3 (antisymmetric: T_{ij} = -T_{ji})
6 + 3 = 9 = dim(V (x) V). Every 2-tensor splits sym + antisym.Vectors versus covectors
An upper-index object v^i lives in V; a lower-index object a_i lives in the dual space V* of linear functionals. Without extra structure these are different spaces — there is no canonical way to turn a vector into a functional. But an inner product supplies exactly that bridge.
Given an inner product with matrix g_{ij}, lowering an index means v_i = g_{ij} v^j (a contraction with the metric), and raising uses the inverse g^{ij}: v^i = g^{ij} v_j. In an orthonormal basis g is the identity, so raising and lowering do nothing visible — which is why Vol I could blur the distinction between a vector and its transpose.
A worked index dance
Let us push indices on a concrete example. Take V = R^2 with the non-orthonormal inner product given by g = [2, 1; 1, 1], so g^{-1} = [1, -1; -1, 2]. We will lower then raise the same vector and confirm we return to where we started.
Metric and its inverse:
g = [ 2 1 ; 1 1 ] g^{-1} = [ 1 -1 ; -1 2 ]
(check: g * g^{-1} = I)
Start with vector v^j = (3, 4) (upper index, a genuine vector).
LOWER: v_i = g_{ij} v^j
v_1 = 2*3 + 1*4 = 10
v_2 = 1*3 + 1*4 = 7 -> v_i = (10, 7) (now a covector)
RAISE back: v^i = g^{ij} v_j
v^1 = 1*10 + (-1)*7 = 3
v^2 = (-1)*10 + 2*7 = 4 -> v^i = (3, 4) back to start.
Norm via the metric (a double contraction):
||v||^2 = g_{ij} v^i v^j = v_i v^i = 10*3 + 7*4 = 58.