Treat coordinates as descriptions of a space

Try

A polynomial can be a vector even though it is not drawn as an arrow. Consider p(t)=2−3t+t² in P₂, the real polynomials of degree at most two. In the ordered basis (1,t,t²), its coordinate column is (2,−3,1). Rewrite the same polynomial in the basis (1,t−1,(t−1)²). Expanding gives p(t)=0−(t−1)+(t−1)², so the new coordinate column is (0,−1,1). The polynomial did not change; its description did.

Predict

Predict whether differentiation on P₂ is invertible. It sends 1 to zero, t to 1, and t² to 2t. Two inputs differing by a constant produce the same derivative, and no derivative in this space has a nonzero t² term. These are two different obstructions—failure of injectivity and failure of surjectivity—which coincide for endomorphisms of one finite-dimensional space. Do not infer their equivalence for maps between spaces of different dimensions.

Experiment

Write a three-row table with input basis vectors 1,t,t² and output coordinate columns (0,0,0),(1,0,0),(0,2,0). Assemble these as columns to obtain the matrix [[0,1,0],[0,0,2],[0,0,0]]. Multiply it by (2,−3,1) to get (−3,2,0), the coordinates of −3+2t. Now replace the input by 5+2t−4t² and check both the matrix calculation and direct differentiation. This two-route check catches a mistaken row/column convention.

Understand

A vector space over a field F is a set with addition and scalar multiplication satisfying the vector-space axioms. Here F will be R or C unless another field is stated. A basis is an independent spanning list, and dimension is its length. A linear map T preserves addition and scalar multiplication: T(au+bv)=aT(u)+bT(v). Once values on a basis are specified, exactly one linear map extends them: expand every vector uniquely and apply the same coefficients to the proposed images. Uniqueness of expansion proves well-definedness; the formula proves linearity. This argument explains why arbitrary basis images determine a matrix, without requiring coordinates to be the vectors themselves.

Apply

Use this construction for a space of symmetric two-by-two matrices. A basis consists of the two diagonal unit matrices and the matrix with both off-diagonal entries equal to one. The trace map sends their coordinates (a,b,c) to a+b. Its kernel has basis the diagonal matrix diag(1,−1) and the off-diagonal basis matrix. Check that the kernel has dimension two and the image dimension one. Explain why multiplying two symmetric matrices is not the addition operation of this vector space and need not produce a symmetric matrix.

A basis also supplies a practical equality test: two vectors are equal precisely when every coordinate agrees. In P₂ this uses equality of polynomials, meaning equality for every input, rather than equality at one sampled point. Two different polynomials can have the same value at a particular t. Compare t and t² at t=0 and t=1; neither agreement proves that the polynomials are the same vector.

Continue

The distinction between an object and its coordinates supports every later operator argument. Return to the existing linear-algebra foundation for elimination and change of basis if needed; this unit assumes those skills while extending them to abstract finite-dimensional spaces. Next, rank-nullity will become a structural decomposition rather than only a pivot count. Infinite-dimensional spaces require extra care: a finite matrix no longer represents every operator, and analytic questions such as continuity are additional hypotheses, not consequences of linearity.

Review Linear Algebra I when a prerequisite needs practice. Continue to Split a space without losing its kernel.