Coordinate Systems and Change of Basis in Linear Algebra

Organizational overview

• Assignment 1 is up; submission via Gradescope is required, even though access issues may occur on Canvas. If issues persist, contact the course staff.
• Deadline: 19 September (more than a month away).
• Collaboration: you may work with others, but everyone must write their own solutions; no AI assistance for the final solutions.
• Question zero on the assignment: list everyone you worked with and describe all collaborations and resources used (university requirement).
• Staff can help with understanding material but not provide solutions.
• Midterm: in week six; will cover material from weeks 1–4 (the previous three weeks plus the current week). You will still have opportunities to practice the material from this week and next week’s workshops before the midterm.
• Any questions about logistics or topic coverage can be asked in class.

Foundations for today’s topic

• Recap from last lecture: linear transformations, nullspace (kernel), column space (range), linear independence, span of a set of vectors, and bases of vector spaces.
• Today’s focus: coordinate systems and coordinate functions; transferring between different bases and different spaces; translating problems from an abstract vector space V to the familiar space R^n.
• Goal: to enable solving problems in V by working with coordinates in R^n, via an appropriate basis.

Key concepts and definitions

• Basis and finite-dimensional spaces
  • Let beta = {b1, …, bn} be a basis for a vector space V.
  • For every v ∈ V there exist scalars c1, …, cn with
    $v = \sum<em>{i=1}^n c</em>i b_i.$
  • The representation is unique (Unique Representation Theorem): the c_i are uniquely determined by v when the basis is fixed.
  • The scalars (c1, …, cn) are collected into a coordinate vector
    $[v]<em>{\beta} = \begin{bmatrix} c</em>1 \vdots \ c_n \end{bmatrix} \in \mathbb{R}^n.$
  • The coordinate vector [v]_{\beta} is called the coordinate representation of v with respect to basis \beta.
• Coordinate mappings and coordinate functions
  • The operation that sends v ∈ V to its coordinate vector [v]{\beta} is a map $\text{coord}</em>{\beta}: V \to \mathbb{R}^n, \quad v \mapsto [v]_{\beta}.$
  • This map is a linear transformation and, for a finite basis, is bijective (an isomorphism between V and \mathbb{R}^n).
  • In many texts, this is denoted with a dot in the middle, e.g. $[v]<em>{\beta} = \text{coord}</em>{\beta}(v).$
• The inverse operation: reconstruction from coordinates
  • If you know the coordinate vector [v]{\beta}, you can reconstruct v via $v = \sum</em>{i=1}^n ([v]<em>{\beta})</em>i \, b_i.$
  • This is the inverse of the coordinate map: coordinates determine the vector uniquely and reconstruct the original vector from its coordinates.
• Visual interpretation in \mathbb{R}^n
  • In \mathbb{R}^n with a given basis, the coordinate vector [v]_{\beta} represents a point in \mathbb{R}^n; changing the basis changes the grid/axes, not the underlying point.
  • For example, in \mathbb{R}^2 with the standard basis e1, e2, the coordinates of v = (x, y) are [v]_{\mathcal{E}} = \begin{bmatrix} x \ y \end{bmatrix}.
  • With a different basis, the same geometric point corresponds to different coordinates in \mathbb{R}^2; the axes are tilted or relocated according to the new basis vectors.
• Linear independence and geometric picture
  • Vectors are linearly independent if no vector is a linear combination of the others; geometrically, they do not lie on the same line (in 2D) or plane (in 3D).
  • Visual intuition: independent basis vectors span distinct directions; dependent sets lie on the same line or plane due to a linear relation.
• Finite basis and isomorphism to \mathbb{R}^n
  • If V has a finite basis of n vectors, then V is isomorphic to \mathbb{R}^n.
  • The coordinate map is a bijective linear transformation between V and \mathbb{R}^n; the two spaces behave identically under linear operations (addition, scalar multiplication) when viewed through coordinates.
  • Isomorphism means: there exists a bijective linear map T: V -> W. Here, coord_{\beta}: V -> \mathbb{R}^n is an example (W = \mathbb{R}^n).
• Examples of isomorphic spaces
  • Example spaces that are essentially the same as vector spaces, via isomorphisms:
  • P_3 (polynomials of degree ≤ 3) and \mathbb{R}^4, with an appropriate basis.
  • 2x2 matrices (which have dimension 4) and \mathbb{R}^4.
  • Important caveat: being isomorphic as vector spaces does not mean the objects are identical; they may have different extra structures (e.g., you can evaluate polynomials at a value, which is not a property of the vector space structure alone).
• What changes of basis do
  • Changing basis is performed by a matrix multiplication that maps coordinates from one basis to another.
  • If B = {b1, …, bn} is a basis for V and we express them in the standard coordinates of \mathbb{R}^n, we can form the matrix
    $P<em>{e\leftarrow\beta} = [\,b</em>1\ b<em>2\ \cdots\ b</em>n\,],$
    whose columns are the basis vectors expressed in the standard basis.
  • For any v ∈ V, if [v]{\beta} is the coordinate vector with respect to B, then the coordinates in the standard basis satisfy $[v]</em>{\mathcal{E}} = P<em>{e\leftarrow\beta} [v]</em>{\beta}.$
  • Conversely, the coordinates with respect to the basis B are obtained by
    $[v]<em>{\beta} = P</em>{e\leftarrow\beta}^{-1} [v]<em>{\mathcal{E}},$ where P{e\leftarrow\beta} is invertible (since B is a basis, the columns are independent).
  • The matrix P_{e\leftarrow\beta} is sometimes denoted as the change-of-basis matrix from \beta-coordinates to standard coordinates.

Worked examples

• Example 1: 2D vector with a non-standard basis
  • Basis: \beta = { b1, b2 } with b1 = (1, 0), b2 = (1, 2) in \mathbb{R}^2.
  • Find coordinates of v = (1, 6) with respect to \beta.
  • Solve for a, b in
    $a b<em>1 + b b</em>2 = v \quad\Rightarrow\quad a(1,0) + b(1,2) = (1,6).$
  • This yields the linear system
    $\begin{cases} a + b = 1 \ 2b = 6 \end{cases} \Rightarrow b = 3, a = -2.$
  • Therefore, [v]_{\beta} = \begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} -2 \ 3 \end{bmatrix}.
• Example 2: Polynomials and the basis {1, t, t^2} in P_2
  • Basis: \beta = {1, t, t^2} for P_2 (polynomials of degree ≤ 2).
  • Coefficients interpretation: a polynomial p(t) = a + b t + c t^2 has coordinates [p]_{\beta} = \begin{bmatrix} a \ b \ c \end{bmatrix}.
  • Given p1(t) = 1 + 2 t^2, [p1]_{\beta} = \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix}.
  • Given p2(t) = 4 + t + 5 t^2, [p2]_{\beta} = \begin{bmatrix} 4 \ 1 \ 5 \end{bmatrix}.
  • If we form the coordinate matrix with columns [p1]{\beta}, [p2]{\beta}, we can compare to a linear combination that yields p_3.
  • Suppose p3(t) is given as a linear combination: p3(t) = -5 p1(t) + 2 p2(t).
    Then in coordinates (with respect to \beta), we have
    [p3]{\beta} = -5 [p1]{\beta} + 2 [p2]{\beta} = -5 \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix} + 2 \begin{bmatrix} 4 \ 1 \ 5 \end{bmatrix} = \begin{bmatrix} 3 \ 2 \ 0 \end{bmatrix}.$n
  • Reconstructing p3 from coordinates: p3(t) = 3 \cdot 1 + 2 t + 0 t^2 = 3 + 2 t.
  • Conclusion: The coordinate vectors of p1, p2, p3 with respect to \beta satisfy the same linear relation as the polynomials themselves in P2, illustrating the correspondence of linear relations under coordinate changes.
• Example 3: Verification of independence using coordinates
  • Take three polynomials p1, p2, p3 in P2, and choose \beta = {1, t, t^2}.
  • Suppose [p1]{\beta} = \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix}, [p2]{\beta} = \begin{bmatrix} 4 \ 1 \ 5 \end{bmatrix}, [p3]{\beta} = \begin{bmatrix} 3 \ 2 \ 0 \end{bmatrix}.
  • Form the 3x3 matrix with these columns and row-reduce to check if the columns are linearly independent.
  • If column 3 is a linear combination of columns 1 and 2 (as above: [p3]{\beta} = -5 [p1]{\beta} + 2 [p2]{\beta}), then p3 is not independent of p1 and p2 in P2; i.e., the three polynomials are linearly dependent in P_2.
  • This mirrors the fact that the column relation in the coordinate space reflects a polynomial relation in the original space.
• Change of basis in practice: a concrete workflow
  • Given a vector v in V and a basis \beta = {b1, …, bn}, compute its coordinates [v]{\beta} by solving $v = \sum</em>{i=1}^n ([v]<em>{\beta})</em>i \, b_i.$
  • To convert to standard coordinates, form the matrix P{e\leftarrow\beta} = [b1 \ b2 \ \cdots \ bn], and compute
    $[v]<em>{\mathcal{E}} = P</em>{e\leftarrow\beta} [v]_{\beta}.$
  • To go from standard coordinates back to \beta-coordinates, use
    $[v]<em>{\beta} = P</em>{e\leftarrow\beta}^{-1} [v]_{\mathcal{E}}.$
  • This framework gives a clean, linear-algebraic way to translate problems into and out of the familiar coordinate space \mathbb{R}^n.

Relationships between abstract spaces and coordinate spaces

• Isomorphism and renaming viewpoint
  • If V has a finite basis, there exists an isomorphism between V and \mathbb{R}^n given by the coordinate map and its inverse.
  • Conceptually, this means you can view V as the same object as \mathbb{R}^n, up to a relabeling of elements (renaming). In particular, vector space operations are preserved under the isomorphism.
  • Example intuition: P_3, 2x2 matrices, and \mathbb{R}^4 are all isomorphic as vector spaces because they all have dimension 4; you can set up a bijective linear map between them preserving addition and scalar multiplication.
• Caveats about isomorphism and extra structure
  • Being isomorphic as vector spaces does not imply all structures are identical. For example, evaluating a polynomial at a value is an operation that is not a property of the vector space itself (it depends on the polynomial structure), whereas the vector space operations are purely linear.
• Practical consequence
  • Properties that are linear in nature (e.g., linear independence, spanning sets, bases) are preserved under the coordinate transformation. If a set of vectors is linearly independent in V, their coordinate vectors in \mathbb{R}^n with respect to some basis are linearly independent in \mathbb{R}^n, and vice versa.
  • If vectors in V span V, then their coordinate vectors span \mathbb{R}^n, and if they form a basis in V, their coordinates form a basis in \mathbb{R}^n.

Selected notes on terminology and notation

• Coordinate notation
  • For v ∈ V with basis \beta, the coordinates are written [v]{\beta} = (c1, …, c_n)^T.
  • The process of converting a vector to its coordinates is a function/linear map coord_{\beta}: V -> \mathbb{R}^n.
• Matrix form of a change of basis
  • If the basis vectors are written in standard coordinates as columns of a matrix P{e\leftarrow\beta}, then for a coordinate vector [v]{\beta}, the standard coordinate vector is
    $[v]<em>{\mathcal{E}} = P</em>{e\leftarrow\beta} [v]_{\beta}.$
  • Inverse relation: $[v]<em>{\beta} = P</em>{e\leftarrow\beta}^{-1} [v]_{\mathcal{E}}.$
• Conceptual takeaway
  • The abstract vector space V and the coordinate space \mathbb{R}^n encode the same linear structure; choosing a basis provides a concrete coordinate system in which to perform calculations, after which results can be translated back to the original space.

Mini-quiz prompts (to test understanding)

• If v ∈ V has basis \beta = {b1, b2, …, bn}, what does the coordinate vector [v]{\beta} represent geometrically?
  • Answer: It represents the unique set of coefficients (c1, …, cn) such that v = \sum{i=1}^n ci b_i; equivalently, it's the coordinates of v in the coordinate system defined by the basis \beta.
• How would you compute the coordinates of a vector in a new basis using a matrix?
  • Answer: Form P{e\leftarrow\beta} with the basis vectors bi as columns in standard coordinates; then [v]{\mathcal{E}} = P{e\leftarrow\beta} [v]{\beta}, so [v]{\beta} = P{e\leftarrow\beta}^{-1} [v]{\mathcal{E}}.
• True/false: If p1, p2, p3 ∈ P2 have coordinate vectors [p1]{\beta}, [p2]{\beta}, [p3]{\beta} that are linearly dependent, then p1, p2, p3 are linearly dependent in P2.
  • Answer: True, because linear relations among coordinate vectors correspond to linear relations among the original vectors under the coordinate map.

Summary

• You can translate problems from an abstract vector space V to a concrete coordinate space \mathbb{R}^n by choosing a basis \beta for V.
• The unique representation of any vector in V with respect to a basis gives a coordinate vector in \mathbb{R}^n.
• The coordinate mapping is linear and bijective, providing an isomorphism between V and \mathbb{R}^n when the basis is finite.
• Change of basis is accomplished via a matrix whose columns are the basis vectors in standard coordinates; this matrix translates coordinates from one basis to another through multiplication (and its inverse).
• These ideas underpin powerful techniques for solving problems in abstract spaces by working entirely in the familiar space \mathbb{R}^n, then translating results back.