Vectors Beyond Coordinates
Transcend the introductory view of vectors as mere lists of numbers or arrows in space. Discover how vectors operate as invariant geometric objects, functional duals, and differential derivations across curved and infinite-dimensional domains.
Like Duolingo, but for Vectors Beyond Coordinates. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
A short one: 11 levels across 2 sections, about 22 minutes end to end, roughly 4 days at five minutes a day. It moves through Freeing the Vector from Coordinates and Vectors in Curved and Infinite Spaces.
Free forever · No credit card · iPhone & Android
Key ideas in Vectors Beyond Coordinates
- A vector exists independently of the coordinate basis used to measure it
- Basis rotations modify coordinate components while leaving inner products and vector magnitude invariant
- Geometric operations like dot products evaluate to identical scalar values across all valid orthonormal bases
- Vector components transform via the inverse of the matrix that transforms the basis vectors (contravariance)
- Scaling a basis up by a factor k scales the components down by 1/k to keep the linear combination sum(v^i * e_i) unchanged
- The opposition between basis transformation and component transformation is mandatory to preserve the physical object
- A vector is zero if and only if all its components vanish, which is invariant across all invertible linear basis transformations
- Mistaking coordinate components for intrinsic objects leads to confusing frame-dependent values with invariant geometric truths, though linear nullity is an invariant property
- Physical laws formulated as vector equations hold across frames precisely because zeroing a geometric vector does not depend on coordinate selection
- Linear operators represent basis-independent endomorphisms whose matrix representations undergo similarity transformations under basis change
- Vector components require the inverse matrix P⁻¹ to compensate for basis re-scaling and rotation
- Matrix similarity preserves spectrum, trace, and determinant precisely because the underlying operator is unchanged by coordinate shifts
- Vector components transform contravariantly via M^-1, whereas covector components transform covariantly via M to preserve scalar evaluation
- A covector is intrinsically an element of Hom(V, F), consuming a vector to yield a scalar, whereas transposition is a basis-dependent matrix convention
- A covector is visually represented by a stack of parallel hyperplanes (level sets of constant value)
- Multiplying a covector by a scalar compresses the spacing between the hyperplanes by that factor
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Vectors Beyond Coordinates the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
Imagine you rotate your coordinate axes by 45 degrees beneath a fixed arrow. What happens to the arrow's length and its numerical components?
Get it right to open this lesson and 10 more in the app.
Where Vectors Beyond Coordinates takes you
- 1
Freeing the Vector from Coordinates
- Invariance and Transformation Laws
- Covectors and the Dual Space
- 2
Vectors in Curved and Infinite Spaces
- Tangent Vectors as Derivations
- Vectors Across Infinite Dimensions
2 sections · 4 units · 11 levels. Built to play, not to enroll.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Vectors Beyond Coordinates is taught in the The Storyteller style: every lesson is a story. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Vectors Beyond Coordinates today.
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