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Quiz

7.1-05 Shears and missing eigen-directions

Shears and missing eigen-directions Explore how a shear can be invertible while providing too few eigen-directions for a basis. Coordinate calculations connect the familiar slanted-grid picture to a repeated eigenvalue and a deficient eigenspace. What you will practise Find the fixed line of a shear Solve the shifted homogeneous equation Compare a nontrivial shear with the identity Track repeated shear applications Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review coordinate transformations and elementary homogeneous systems. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final reasoning task links the missing independent direction to failure of diagonalization. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.1-04 Rotations and real eigenvectors

Rotations and real eigenvectors Investigate when a rotation preserves a real line and when complex eigenvalues are needed. The examples include planar rotations, spatial axes and the important zero-angle and half-turn exceptions. What you will practise Analyze planar rotation exceptions Identify a spatial rotation axis Use length preservation to constrain eigenvalues Explain the field dependence of eigendirections Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review planar and spatial geometry, basic trigonometry and eigenvectors. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. A four-dimensional example tests whether a familiar three-dimensional fixed-axis intuition really generalizes. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.1-03 Reflections and projections

Reflections and projections Use the geometry of reflections and projections to identify eigenspaces before computing a characteristic polynomial. Move from coordinate-axis examples to planes, oblique projections and a reflection built from a projection. What you will practise Identify fixed and reversed directions Find kernels of projections Count eigenspace dimensions geometrically Connect projection identities to eigenvalues Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review subspaces, perpendicular vectors and basic matrix multiplication. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final task connects a projection to a reflection and checks the result on complementary subspaces. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.1-02 Positive, negative and zero eigenvalues

Positive, negative and zero eigenvalues Interpret the sign and magnitude of a real eigenvalue as two separate pieces of geometric information. Follow stretching, reversal and collapse through one or several applications of a transformation. What you will practise Separate direction from length Interpret zero eigenvalues Track even and odd iterations Combine independent coordinate directions Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review scalar multiplication and the definition of an eigenvector. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The later exercises combine different multipliers in a single initial vector rather than treating every vector as one eigenmode. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

7.1-01 Invariant directions

Invariant directions Connect eigenvectors to lines that a linear transformation preserves. Work from geometric language to component checks and small parameter problems, distinguishing a direction from a particular arrow on it. What you will practise Interpret the eigenvector equation Check a candidate using matrix multiplication Recognize nonzero scalar multiples Reason about invariant lines and sums Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix–vector multiplication and scalar multiples. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. A final parameter problem requires the same eigenvalue to satisfy every coordinate. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

linear algebra

Oct 20, 2026

8 questions

Quiz

6.6-10 Finite projections and infinite series

Finite projections and infinite series Finite projection geometry motivates Fourier series but does not establish every infinite-limit claim. This final microlearning separates mean-square, pointwise and uniform convergence and identifies the role of completeness and regularity assumptions. What you will practise State mean-square convergence precisely. Use nested projection spaces without overclaiming. Distinguish orthonormality from completeness. Interpret point changes, counterexamples and conditional jump convergence. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review finite Fourier projections and basic sequences of functions. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports careful study of Fourier and Hilbert-space convergence theory within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

8 questions

Quiz

6.6-09 Fourier coefficients as projections

Fourier coefficients as projections Finite Fourier approximations are projections onto selected sine and cosine directions. The set states its coefficient convention explicitly, then uses orthogonality, symmetry and energy calculations to determine finite approximations. What you will practise Track the constant-term normalization convention. Calculate sine and cosine coefficients. Use parity and known trigonometric expansions. Evaluate the error from omitted frequencies. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review trigonometric integrals and function-space projection. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports understanding Fourier coefficients before studying infinite-series convergence within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

10 questions

Quiz

6.6-08 Best polynomial approximation

Best polynomial approximation Best polynomial approximation projects a function onto a finite polynomial space. These questions combine concrete low-degree approximations with residual orthogonality, error calculations and the distinction between integral fitting and interpolation. What you will practise Form projection coefficients and function-space Gram equations. Approximate polynomial targets by lower-degree spaces. Calculate minimum squared integral error. Reason about nested spaces and uniqueness. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review orthogonal polynomials and the closest-point theorem. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports finite Fourier projection and broader approximation methods within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

12 questions

Quiz

6.6-07 Building orthogonal polynomials

Building orthogonal polynomials Build low-degree orthogonal polynomials by applying Gram–Schmidt to monomials. Practice uses the unweighted integral product on the symmetric unit interval and distinguishes monic, endpoint and unit-norm normalizations. What you will practise Construct constant, linear and quadratic orthogonal directions. Calculate projection coefficients and squared norms. Relate the results to low-degree Legendre polynomials. Extend the construction to a cubic and explain dependence on the inner product. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review Gram–Schmidt and polynomial product integrals. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports best polynomial approximation in an orthogonal basis within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

12 questions

Quiz

6.6-06 Inner products of functions

Inner products of functions Integration turns continuous functions into geometric objects with length and perpendicularity. The questions calculate products and distances, test orthogonality on specified intervals and explain why integral geometry differs from pointwise overlap. What you will practise Evaluate polynomial function products. Compute integral norms and distances. Use parity and centered functions. Distinguish continuous-function positivity from sampled degeneracy. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review the integration bridge and real inner-product axioms. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports orthogonal polynomials and approximation of functions within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

10 questions

Quiz

6.6-05 Frobenius inner products

Frobenius inner products Matrices themselves form an inner product space when corresponding entries are paired. This questionset links the Frobenius entrywise definition with traces, norms, orthonormal matrix units and projection onto a matrix subspace. What you will practise Compute Frobenius products and distances. Use the correct trace identity. Recognize orthogonal matrix subspaces. Check invariance under compatible orthogonal transformations. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review matrix entries, transposes and inner-product projection. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports matrix approximation and later low-rank methods within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

8 questions

Quiz

6.6-04 Weighted inner products

Weighted inner products A positive-definite weight matrix gives a new geometry on familiar coordinate vectors. Practice checks valid weights, computes products and projections, and uses coordinate transformations to understand weighted length and perpendicularity. What you will practise Test symmetric positive-definite weight matrices. Calculate weighted products and perpendicular vectors. Interpret transformed Euclidean geometry. Analyze weighted projection and parameter boundaries. Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail. Before you start Review inner products, quadratic forms and basic matrix algebra. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports weighted fitting and geometric changes of coordinates within Module 6 of the Complete Linear Algebra Course.

linear algebra

Oct 20, 2026

10 questions