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Math101learn.math101.caBasis
A rigorous guide to bases, coordinates, change of representation, and proof of spanning plus independence.
Precise definition
A basis of a vector space $V$ is an ordered list $(v_1,\ldots,v_n)$ that is linearly independent and spans $V$. Every $v\in V$ then has a unique representation $v=c_1v_1+\cdots+c_nv_n$. The coordinate vector is $[v]_B=(c_1,\ldots,c_n)^T$ and depends on the chosen ordered basis $B$.
Notation and mathematical language
The standard basis of $\mathbb R^n$ is $(e_1,\ldots,e_n)$. A change-of-coordinates matrix $P_{C\leftarrow B}$ has columns $[b_j]_C$ and satisfies $[v]_C=P_{C\leftarrow B}[v]_B$. Reordering a basis changes coordinates even though the underlying vectors and space do not change.
Conceptual picture
Spanning ensures enough directions; independence removes redundancy. Uniqueness follows because two coordinate representations subtract to a linear relation among basis vectors, forcing all coefficient differences to zero.
Conditions and key results
For a finite-dimensional space of dimension $n$, any independent list of $n$ vectors is automatically a basis, as is any spanning list of $n$ vectors. In an abstract space, vectors may be polynomials, matrices, or functions, and row reduction must be applied to their coordinate representations in a known ambient basis.
A reliable strategy
- Identify the vector space, field, and its known or expected dimension.
- Arrange candidate coordinate vectors as columns and solve the homogeneous dependence equation.
- Verify independence and spanning, or use the dimension theorem once one property and the correct count are established.
- Solve for coordinates, preserve basis order, and reconstruct the vector as a check.
Fully worked example
Interpretation and application
Bases compress a vector into coordinates suited to a problem: Fourier modes, eigenvectors, polynomial coefficients, or geometric axes. Coordinates change with representation, while statements such as linear independence and dimension are intrinsic.
Common mistakes
Verification and reasonableness
- Reconstruct the original vector from its computed coordinates.
- Check the column matrix has a pivot in every column and every ambient row when dimensions match.
- Verify that the zero vector has only the zero coordinate list.
Practice
- Is $((1,0),(2,0))$ a basis of $\mathbb R^2$?
- Find $[(4,2)]_B$ for $B=((1,1),(1,-1))$.
- How many vectors are in a basis of a four-dimensional space?
Answers and brief solutions
- No; the vectors are dependent and span only the $x$-axis.
- $(3,1)^T$.
- Exactly four.
Further deduction
The exchange lemma explains basis replacement: if $B$ is a basis and a nonzero vector $w$ has a nonzero coefficient on $b_k$, then replacing $b_k$ by $w$ still gives a basis. This supports algorithms that extend independent sets or trim spanning sets and proves that all finite bases of a space have the same size.
Coordinate change is reversible: $P_{B\leftarrow C}=(P_{C\leftarrow B})^{-1}$. If a linear transformation has matrix $A_B$ in basis $B$ and $P=P_{B\leftarrow C}$ converts $C$-coordinates to $B$-coordinates, then $A_C=P^{-1}A_BP$. This similarity relation explains why trace, determinant, and eigenvalues do not depend on the chosen basis even though entries do.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For $B=((1,1),(1,-1))$, what is the first coordinate of $(6,2)$?
- Adding the two equations gives $2c_1=8$.
- Thus $c_1=4$.
End of lesson
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