piatra . institutepathwaysFinding the next step…
Build geometric and algebraic understanding of vector spaces, linear maps, eigenvectors and decompositions.
No subject-specific background required; work through the opening foundation.
A least-squares model explained both geometrically and computationally.
Dan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingLinear algebra: systems. Interpret a system geometrically.
Linear algebra: row reduction. Show which operations preserve solutions.
Dan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingLinear algebra: vectors. Connect coordinates with vector operations.
Linear algebra: vector equations. Translate a system into a span question.
Linear algebra: solution sets. Find what combinations can reach.
How many solutions satisfy x + y = 3 and 2x + 2y = 6? Explain without treating them as two independent constraints.
Compare the second equation with twice the first.
Infinitely many. The second equation is twice the first and adds no new constraint. Every pair (x, y) = (t, 3 − t) satisfies both. Geometrically the equations describe the same line.
What to look for
Common mistake: Counting equations instead of checking whether their constraints are independent.
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Dan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
Check reading access
Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingLinear algebra: matrix equations. Trace dimensions in Ax=b.
Linear algebra: matrix transformations. Draw the images of basis vectors.
Linear algebra: projections. Decompose a vector into projection and residual.
Linear algebra: least squares. Derive the normal equations and inspect their limits.
A = [[2, 0], [0, 1]] and v = (3, 2). Calculate Av. Does this transformation preserve every vector’s length?
Look at the effect on each coordinate or on the basis vector (1, 0).
Av = (6, 2). The transformation doubles the first coordinate and preserves the second. It does not preserve every length: (1, 0) becomes (2, 0), changing its length from 1 to 2.
What to look for
Common mistake: Assuming a matrix transformation must be a rotation or preserve length.
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Dan Margalit · Joseph Rabinoff · reference
Named section and worked examples
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Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
Check reading access
Open the readingDan Margalit · Joseph Rabinoff · reference
Named section and worked examples
Check reading access
Open the readingLinear algebra: linear independence. Identify redundant directions.
Linear algebra: subspaces. Check closure rather than visual resemblance.
Linear algebra: basis and dimension. Explain why coordinate descriptions need a basis.
You approximate the observations 1, 2 and 6 by one constant c. Which c minimises the sum of squared residuals? Show why.
Expand (1 − c)² + (2 − c)² + (6 − c)² or consider the mean.
The best constant is c = 3, the mean. The squared-error sum is 3c² − 18c + 41 = 3(c − 3)² + 14. It is minimised at c = 3 with error 14. The residuals are −2, −1 and 3, whose sum is zero, although none of the observations has to equal the fitted value.
What to look for
Common mistake: Choosing the median or assuming a fit is invalid whenever residuals remain.
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Editorial perspectives based on selected works, rather than author-endorsed reading lists.
An editorial comparison of the core argument, alternative explanations and practical evidence.
A curated synthesis cannot be neutral or exhaustive. Follow the primary sources when an interpretation matters.
An editorial route that starts with working examples.
The order supports a particular study method; it does not establish which account is true.
An editorial route that starts with competing explanations.
The order supports a particular study method; it does not establish which account is true.
Background, different viewpoints, and further reading.
Interpret a system geometrically.
Named section and worked examples
Show which operations preserve solutions.
Named section and worked examples
Describe the whole solution set.
Named section and worked examples
Connect coordinates with vector operations.
Named section and worked examples
Translate a system into a span question.
Named section and worked examples
Find what combinations can reach.
Named section and worked examples
Identify redundant directions.
Named section and worked examples
Check closure rather than visual resemblance.
Named section and worked examples
Explain why coordinate descriptions need a basis.
Named section and worked examples
Trace dimensions in Ax=b.
Named section and worked examples
Draw the images of basis vectors.
Named section and worked examples
Test additivity and scalar compatibility.
Named section and worked examples
Interpret multiplication as composition.
Named section and worked examples
Distinguish solving from explicitly computing an inverse.
Named section and worked examples
Interpret determinant as signed volume scaling.
Named section and worked examples
Find invariant directions of a map.
Named section and worked examples
Explain when an eigenbasis exists.
Named section and worked examples
Connect dot products and perpendicularity.
Named section and worked examples
Decompose a vector into projection and residual.
Named section and worked examples
Derive the normal equations and inspect their limits.
Named section and worked examples
Use vectors, matrix operations and geometric intuition to explain a model’s calculations.
Use vectors and operators in the worked quantum and Bell-type calculations.
Matrix notation supports the discrete models. It complements probability preparation rather than replacing it.
Compare geometric models with explanations of continuous change.