Key Concepts of Basis Vectors

Why This Matters

Basis vectors are the backbone of everything you'll do in linear algebra—from solving systems of equations to understanding matrix transformations and eigenvalue problems. When you grasp how basis vectors work, you're not just learning definitions; you're building the mental framework for span, linear independence, dimension, and coordinate representation. These concepts appear repeatedly throughout the course, and exam questions will test whether you understand the underlying structure, not just the formulas.

Here's the key insight: basis vectors give us a "coordinate language" for talking about any vector in a space. Every topic that follows—change of basis, orthogonalization, diagonalization—builds on this foundation. So don't just memorize that basis vectors must be linearly independent and span the space. Know why those two properties matter and how they connect to dimension, unique representation, and transformations.

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Foundational Definitions and Properties

A basis is the minimal spanning set for a vector space—it has exactly enough vectors to reach everywhere, with no redundancy.

Definition of Basis Vectors

• A basis is a set of vectors that spans the entire vector space—meaning any vector in the space can be written as a linear combination of basis vectors
• Linear independence is required—no basis vector can be expressed as a combination of the others, ensuring no redundancy
• Coordinates arise from the basis—once you fix a basis, every vector has a unique representation as coefficients of the basis vectors

Properties of Basis Vectors

• The number of basis vectors equals the dimension—this is the fundamental link between bases and the "size" of a vector space
• Unique representation guaranteed—every vector in the space corresponds to exactly one set of coefficients relative to the basis
• All bases for a space have the same size—you can swap out which vectors you use, but the count stays fixed

Standard Basis Vectors

• Standard basis vectors are unit vectors along coordinate axes—in $\mathbb{R}^n$, these are $\mathbf{e}_1 = (1,0,\ldots,0)$, $\mathbf{e}_2 = (0,1,\ldots,0)$, etc.
• They provide the "default" coordinate system—when no basis is specified, assume the standard basis
• Simplest for computation—coordinates of a vector relative to the standard basis are just its components

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Linear Independence and Span

These two properties are the "tests" a set of vectors must pass to qualify as a basis—span ensures coverage, independence ensures efficiency.

Linear Independence of Basis Vectors

• A set is linearly independent if the only solution to $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n = \mathbf{0}$ is all coefficients equal to zero—this is the formal test
• Dependent sets contain redundancy—at least one vector can be removed without shrinking the span
• Independence guarantees unique coordinates—without it, the same vector could have multiple representations

Span of Basis Vectors

• Span is the set of all linear combinations—written $\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}$, it's every vector you can "reach" using those vectors
• A basis must span the entire space—if the span is smaller than the full space, you need more vectors
• Span can be visualized geometrically—two non-parallel vectors in $\mathbb{R}^3$ span a plane, not the full space

Dimension of a Vector Space

• Dimension equals the number of vectors in any basis—this is a theorem, not just a definition
• Dimension tells you degrees of freedom—in $\mathbb{R}^3$, you need exactly 3 coordinates to specify a point
• Subspaces have smaller dimension—a plane through the origin in $\mathbb{R}^3$ is a 2-dimensional subspace

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Changing and Constructing Bases

Different bases reveal different structure in the same space—choosing the right basis can dramatically simplify a problem.

Change of Basis

• Change of basis expresses vectors using different "coordinates"—the vector itself doesn't change, only its representation
• A change of basis matrix $P$ converts coordinates—if $[\mathbf{v}]_B$ are coordinates in basis $B$, then $[\mathbf{v}]_{B'} = P^{-1}[\mathbf{v}]_B$
• Strategic basis choice simplifies problems—diagonal matrices, for instance, are much easier to work with than dense ones

Orthonormal Basis

• Orthonormal means orthogonal (perpendicular) and normalized (unit length)—formally, $\mathbf{u}_i \cdot \mathbf{u}_j = 0$ for $i \neq j$ and $\|\mathbf{u}_i\| = 1$
• Projections become simple dot products—the coordinate of $\mathbf{v}$ along $\mathbf{u}_i$ is just $\mathbf{v} \cdot \mathbf{u}_i$
• The change of basis matrix is orthogonal—meaning $P^{-1} = P^T$, which makes computations fast and stable

Gram-Schmidt Process

• Gram-Schmidt converts any basis to an orthonormal one—it systematically removes the "overlap" between vectors
• The algorithm: orthogonalize, then normalize—subtract projections onto previous vectors, then scale to unit length
• Preserves the span at each step—the new orthonormal basis spans exactly the same space as the original

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Bases in Transformations and Applications

The power of basis vectors becomes clear when you see how they interact with linear transformations—the basis you choose determines how "nice" your matrix looks.

Basis Vectors in Matrix Transformations

• A matrix's columns show where basis vectors land—if $A$ is a transformation matrix, column $j$ is $A\mathbf{e}_j$
• Understanding the action on basis vectors reveals the full transformation—linearity means everything else follows
• Different bases yield different matrix representations—the same transformation can look simple or complicated depending on your choice

Eigenvectors as Basis Vectors

• Eigenvectors satisfy $A\mathbf{v} = \lambda\mathbf{v}$—they only get scaled, not rotated, by the transformation
• An eigenbasis diagonalizes the matrix—if you can form a basis of eigenvectors, the matrix becomes diagonal in that basis
• Diagonalization simplifies powers and exponentials—computing $A^{100}$ is trivial when $A$ is diagonal

Basis Vectors in Coordinate Systems

• The choice of basis defines your coordinate system—Cartesian, polar, and other systems correspond to different basis choices
• Non-orthogonal bases are valid but messier—coordinates still exist, but formulas for length and angle become more complex
• Physical applications often suggest natural bases—principal axes, normal modes, and other structures guide basis selection

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Quick Reference Table

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Self-Check Questions

1. What two properties must a set of vectors satisfy to be a basis, and why is each property necessary?

2. If you have 4 vectors in $\mathbb{R}^3$, can they form a basis? Explain using the concept of dimension.

3. Compare and contrast the standard basis with an eigenbasis for a matrix $A$. When would you prefer each?

4. Describe how the Gram-Schmidt process transforms a basis. What properties does the output have that the input might lack?

5. If a matrix $A$ acts on a vector $\mathbf{v}$, how can you determine $A\mathbf{v}$ by only knowing what $A$ does to the basis vectors? Why does this work?