What is the difference between a vector and a covector in tensor calculus?

A vector represents a geometric direction or displacement, while a covector is a linear function that takes a vector as input and returns a scalar.

How do tensor components transform when the coordinate basis changes?

Contravariant vector components transform with the inverse basis transformation, while covector components transform with the forward transformation. General tensors follow the corresponding rule for each index.

How does the metric tensor raise and lower indices?

The metric tensor lowers vector indices via v_i = g_ij v^j, and its inverse raises covector indices via v^i = g^ij v_j.

Free Course Image Tensors for Beginners: Learn Tensor Calculus Fundamentals

Free online course Tensors for Beginners: Learn Tensor Calculus Fundamentals

Duration of the online course: 3 hours and 6 minutes

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Build intuition for tensor calculus fast with a free online course: vectors, covectors, metrics, and index notation explained clearly, plus practice questions.

In this free course, learn about

Motivation for tensors (e.g., geometry/physics) and why coordinates alone can mislead
Coordinate-free vs coordinate (component) definition of tensors and what makes an object a tensor
Forward vs backward change-of-basis matrices and their inverse relationship
Vector components vs basis vectors; how vector components transform under basis scaling/change
What covectors (dual vectors) are and how dual basis covectors pick out vector components
Covector component and basis transformation rules (contravariant vs covariant behavior)
Linear maps as (1,1)-tensors; how their components transform between bases
Metric tensor in non-orthonormal coordinates to compute lengths/dot products correctly
Bilinear forms as (0,2)-tensors; extra properties needed to be a metric (symmetric, nondegenerate)
Linear maps as vector–covector pairs; meaning of “pure” vs “impure” matrix columns
Bilinear forms as covector–covector pairs and why this naturally encodes bilinearity
Tensor product vs Kronecker product: abstract multilinear construction vs matrix layout operation
General tensors as multilinear maps on vectors/covectors; ambiguity of naive notation for higher types
Tensor product spaces; raising/lowering indices via the metric (flat/sharp operators)

About the free online course

This free online course helps you build a clear, usable understanding of tensor calculus fundamentals without getting lost in heavy formalism. If you have met vectors in calculus or physics and wondered why tensors appear in topics like changing coordinates, curved spaces, or advanced mechanics, this course connects the motivation to practical definitions that actually work when you compute. You will learn to recognize what stays the same under a change of basis and what changes, which is the central idea behind tensors.

Instead of treating tensors as mysterious arrays, you develop intuition step by step: starting from coordinate descriptions, then moving through vectors and their transformation rules, and on to covectors as objects that eat vectors and return numbers. That perspective makes later ideas feel natural, because you can see how dual bases, components, and transformations fit together. Along the way, you practice distinguishing forward and backward transformations so you can reliably translate between old and new coordinate systems.

As the course progresses, tensors appear as structured combinations of vector and covector behavior. You explore linear maps as (1,1)-tensors and understand how their components transform, which clarifies when a matrix representation is tied to a specific basis. You then meet bilinear forms and the metric tensor, learning why they matter for computing lengths and angles correctly in non-orthonormal coordinates and for moving between upstairs and downstairs indices through raising and lowering operations.

You will also gain a conceptual handle on tensor products, how they differ from the Kronecker product in purpose and meaning, and why higher-type tensors can become ambiguous if you only think in terms of symbols rather than multilinear maps. Short exercises throughout reinforce the transformation rules and definitions so you finish with confidence reading index notation, checking whether an expression is coordinate-consistent, and setting yourself up for applications in multivariable calculus, differential geometry, physics, computer graphics, and machine learning.

Course content

Video class: Tensors For Beginners (-1): Motivation 06m
Exercise: Which mathematical topic is presented as the main motivation for studying tensors?
Video class: Tensors for Beginners 0: Tensor Definition 09m
Exercise: Which statement best matches the “coordinate definition” of a tensor?
Video class: Tensors for Beginners 1: Forward and Backward Transformations (REMAKE) 11m
Exercise: What is the relationship between the forward transformation matrix F and the backward transformation matrix B?

Video class: Tensors for Beginners 2: Vector definition 09m
Video class: Tensors for Beginners 3: Vector Transformation Rules 05m
Exercise: If the basis vectors are scaled up by a factor of 2, how do the components of a fixed vector change in the new basis?

Video class: Tensors for Beginners 4: What are Covectors? 14m
Exercise: Which statement best describes a covector?
Video class: Tensors for Beginners 5: Covector Components (Contains diagram error; see description) 08m
Exercise: What does applying the dual basis covector \(\varepsilon^i\) to a vector \(v\) return?
Video class: Tensors for Beginners 6: Covector Transformation Rules 06m
Exercise: Which transformation is used to express the new dual basis covectors in terms of the old dual basis covectors?

Video class: Tensors for Beginners 7: Linear Maps 12m
Video class: Tensors for Beginners 8: Linear Map Transformation Rules 11m
Exercise: How do the components of a linear map (a (1,1)-tensor) transform when changing from an old basis to a new basis?

Video class: Tensors for Beginners 9: The Metric Tensor 16m
Exercise: In a non-orthonormal coordinate system, which object lets you correctly compute the squared length of a vector using its components and basis dot products?
Video class: Tensors for Beginners 10: Bilinear Forms 09m
Exercise: Which additional properties make a bilinear form a valid metric tensor (beyond being a (0,2)-tensor)?

Video class: Tensors for Beginners 11: Linear maps are Vector-Covector Pairs 12m
Exercise: What distinguishes a pure matrix (pure linear map) from an impure one in terms of its columns?
Video class: Tensors for Beginners 12: Bilinear Forms are Covector-Covector pairs 07m
Exercise: Why are covector–covector pairs a natural building block for bilinear forms?
Video class: Tensors for Beginners 13: Tensor Product vs Kronecker Product 04m
Exercise: Which statement best distinguishes the tensor product from the Kronecker product?
Video class: Tensors for Beginners 14: Tensors are general vector/covector combinations 08m
Exercise: Why is the expression Q(D) considered ambiguous for higher-type tensors?
Video class: Tensors for Beginners 15: Tensor Product Spaces 15m
Exercise: What does it mean to say a tensor (viewed as a function) is a multilinear map?

Video class: Tensors for Beginners 16: Raising/Lowering Indexes (with motivation, sharp flat operators) 15m
Exercise: Which object is used to lower indexes (convert a vector’s components from upstairs to downstairs)?