Essense of Linear Algebra by 3Blue1Brown

by Shiju P John

Oct 9, 2025

11 quizzes

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Quizzes covering the famous Youtube playlist "Essense of Linear Algebra", in the channel 3Blue1Brown

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11 quizzes

Showing 1–11 of 11 quizzes · Page 1 of 1

3Blue1Brown: Vectors | Essence of Linear Algebra - Chapter 1 Quiz

Vectors: geometric and numerical reasoning This quiz accompanies Chapter 1 of 3Blue1Brown’s Essence of Linear Algebra. It revisits vector interpretations, coordinates, addition, scaling and applications. Practice Read coordinates in two and three dimensions. Connect arrows with ordered numerical components. Interpret addition as successive displacements. Apply scalar multiplication to every component. For vectors in the same coordinate system, addition follows $$\begin{pmatrix}a\\b\end{pmatrix}+\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a+c\\b+d\end{pmatrix}$$. Scaling follows $$k\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}ka\\kb\end{pmatrix}$$. A negative factor reverses a nonzero vector’s direction. Basic signed arithmetic and Cartesian coordinates are useful prerequisites. Work out the answer before reading the explanation, and compare the numerical calculation with a sketch. The quiz covers introductory vector ideas; it does not require determinants, eigenvalues or matrix inversion.

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3Blue1Brown: Linear Combinations, Span, and Basis Vectors - Chapter 2 Quiz

Linear combinations, span and bases This quiz accompanies Chapter 2 of 3Blue1Brown’s Essence of Linear Algebra. It connects coordinates and geometric reach with the algebra of linear combinations. Study focus Interpret coordinates as coefficients for an ordered basis. Determine whether vectors span a point, line, plane or full space. Recognize redundant vectors and distinguish independence from spanning. Use the basis criterion: independence together with spanning. Calculate coordinates when a different basis is specified. A linear combination has the form $$c_1v_1+\cdots+c_kv_k$$. The span contains every such combination. For vectors in $$\mathbb{R}^n$$, its dimension is at most $$\min(k,n)$$; redundancy can make it smaller. A basis gives each vector one unique coefficient list. Basic vector addition, scalar multiplication and signed arithmetic are useful prerequisites. Some questions require selecting several answers. Check each condition separately, including zero-vector cases, before consulting the explanation. Numeric examples are intended to be worked out rather than recalled. Changing coordinates changes a representation, while the underlying vector remains fixed.

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3Blue1Brown: Linear Transformations and Matrices - Chapter 3 Quiz

Linear transformations and matrices This quiz follows Chapter 3 of 3Blue1Brown’s Essence of Linear Algebra. It practices the connection between movement in the plane and calculations with matrices. Practice topics Recognize linear maps and distinguish nonzero translations and nonlinear coordinate rules. Build a matrix from standard basis images. Compute matrix-vector products and recover input vectors from simple equations. Interpret rotations, reflections, shears, scalings and projections. Relate dependent columns to loss of dimension and invertibility. A linear map satisfies $$L(u+v)=L(u)+L(v)$$ and $$L(cu)=cL(u)$$. With column-vector coordinates, $$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$$. Prior familiarity with vectors, linear combinations and Cartesian coordinates is useful. Some questions require multiple selections. Check every alternative independently and calculate before reading the explanation. Remember that a singular linear map may collapse the plane to a line or a point; visual rules must allow these cases. The quiz introduces geometric interpretation rather than a complete treatment of determinants or eigenvalues.

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3Blue1Brown: Matrix Multiplication as Composition - Chapter 4 Quiz

Matrix multiplication as composition This quiz accompanies Chapter 4 of 3Blue1Brown’s Essence of Linear Algebra. It practices reading products as sequences of linear transformations and connecting algebraic rules with geometric effects. Study focus Read products from right to left when they act on column vectors. Construct product columns by transforming the right-hand matrix’s columns. Distinguish associativity from commutativity. Interpret repeated projections, reflections, identity maps and inverses. Calculate simple composite maps and determinant effects. For compatible matrices, $$ABv=A(Bv)$$ and $$A(BC)=(AB)C$$. The order of factors usually matters, although particular pairs may commute. For square matrices, $$\det(AB)=\det(A)\det(B)$$. Familiarity with vector coordinates, linear combinations and matrix-vector multiplication is recommended. Some items require several selections. Trace basis vectors or calculate a small example before reading the explanation. Pay attention to stated invertibility and nonzero-vector assumptions. An inverse must exist before it can undo a map, and a fixed vector is an eigenvector only when it is nonzero. The quiz includes introductory applications beyond the chapter’s main composition theme.

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3Blue1Brown: 3D Transformations - Chapter 5 Quiz

This quiz reviews three-dimensional linear transformations from Chapter 5 of 3Blue1Brown’s Essence of Linear Algebra. It connects the geometry of transformed space with the entries and columns of a 3 × 3 matrix. Topics include images of the standard basis vectors, matrix–vector multiplication, axis rotations, scaling, reflection, orthogonal projection, and the identity and zero transformations. Questions also explore how linear maps affect the origin and equally spaced points, including maps that collapse dimensions. Rotation questions require attention to the stated axis, viewing direction, and right-hand convention. The composition exercises use column vectors and ask you to connect the order of operations with matrix products. You will also consider why following basis vectors simplifies three-dimensional visualization and how these ideas relate to computer graphics and robotics. Before starting, review coordinates in three dimensions, linear combinations, and basic matrix multiplication. Work out the images of the basis vectors or trace a general input vector to check your reasoning. The explanations support revision after each attempt and distinguish geometric intuition from the algebraic calculation.

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3Blue1Brown: The Determinant - Chapter 6 Quiz

The determinant: area, volume and orientation This quiz follows Chapter 6 of 3Blue1Brown’s Essence of Linear Algebra. It combines geometric interpretation with small determinant calculations. Practice Compute 2×2 determinants and interpret their sign and magnitude. Distinguish signed area or volume from ordinary nonnegative measurements. Recognize dimension collapse and dependent matrix columns. Reason about transformed unit squares and unit cubes. Combine determinant effects of successive transformations. For a square matrix $$\begin{pmatrix}a&b\\c&d\end{pmatrix}$$, the determinant is $$ad-bc$$. Its absolute value scales ordinary area. In three dimensions, the absolute determinant scales ordinary volume. A negative nonzero determinant reverses orientation; zero determinant indicates loss of dimension. Prior knowledge of vector coordinates, basis images and matrix-vector multiplication is useful. Several questions require multiple selections. Check each statement separately, and consider the zero matrix as well as nondegenerate examples. When using parallelogram vertices, identify the ordered basis images before determining a sign. Work through calculations and sketches before reading the explanations.

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3Blue1Brown: Essence of Linear Algebra - Chapter 7 - Inverse matrices, column space and null space - Quiz

Inverse matrices, column space and null space This quiz accompanies Chapter 7 of 3Blue1Brown’s Essence of Linear Algebra. It tests the geometric meaning of matrix transformations and linear systems rather than algorithms for computing inverses. Topics and preparation Review vectors, spans, matrix-vector multiplication and determinants before attempting the questions. The quiz connects invertibility of square matrices with dimension preservation, and examines how column space controls the existence of solutions and null space controls their uniqueness. Interpret the identity and inverse transformations. Distinguish rank, nullity, outputs and inputs mapped to zero. Reason about singular transformations in two and three dimensions. Recognize linear equations and the matrix form of a system. Pay attention to whether a matrix is square: determinant-based criteria apply to square matrices, while column space and null space are defined for rectangular matrices too. Questions include both single-choice and multiple-select items.

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3Blue1Brown: Linear Algebra : Nonsquare Matrix Transformations between dimensions -Chapter 8 Quiz

Nonsquare matrices and transformations between dimensions This quiz accompanies Chapter 8 of 3Blue1Brown’s Essence of Linear Algebra. It explores how rows and columns identify the codomain and domain of a matrix transformation, and how transformed input basis vectors determine its columns. Prerequisites include vectors, linear combinations, spans and basic matrix multiplication. Questions cover concrete matrix construction, rank and nullity, composition, scalar-valued maps and their connection to dot products. Distinguish the codomain from the actual image or column space. Explain why mapping from higher to lower dimensions cannot be injective. Recognize when a map into a larger space is an embedding. Interpret full rank for tall and wide matrices. Check which matrix operations are defined for a given shape. Use geometric intuition alongside the algebraic definition of linearity. A change in domain and codomain dimensions does not change those axioms, and a larger codomain does not guarantee a larger image. Both single-choice and multiple-select questions are included.

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3Blue1Brown: Linear Algebra: Dot Products and Duality - Chapter 9 Quiz

Dot products and duality This quiz connects coordinate arithmetic with Euclidean geometry and scalar-valued linear maps. It accompanies Chapter 9 of 3Blue1Brown’s Essence of Linear Algebra. Review vector coordinates, lengths, angles, projections and matrix-vector multiplication. Questions ask you to calculate dot products, interpret their signs, explain symmetry and scaling, and construct row matrices from basis images. Distinguish signed scalar projection from an unsigned length or vector projection. Relate a Euclidean vector to the linear functional defined by dotting with it. Check how unit and non-unit vectors affect projection interpretation. Identify violations of linearity and handle zero-vector or collapsed-direction cases. The vector–functional correspondence here uses the standard real Euclidean dot product. Signed projections preserve directional information, and projection formulas involving a direction require a nonzero direction vector. Read multiple-select questions carefully and judge each statement independently.

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3Blue1Brown: Linear Algebra Chapter 13: Change of Basis and Coordinate Systems

Change of basis and coordinates Explore the distinction between a geometric vector and the coordinate numbers used to describe it. This questionset follows the linear-coordinate examples in 3Blue1Brown’s change-of-basis lesson, including an alternative ordered basis and its skewed grid. Vector representations Practise reading coordinates as coefficients of basis vectors, building a basis matrix from columns, and translating vectors between standard and alternative coordinates. Work through forward conversion, inverse conversion and reconstruction checks. Linear basis changes share the zero vector; they do not include an independent shift of origin. Representing transformations Understand matrix columns as images of basis vectors, distinguish active geometric maps from passive coordinate changes, and interpret the sequence needed to represent a transformation in another basis. Rotation examples connect abstract matrix composition with concrete vector calculations. Prerequisites are basic matrix-vector multiplication, linear combinations, independent vectors and matrix inverses. Questions use column-vector coordinates. Keep track of each matrix’s input and output basis: the same coordinate pair can describe different vectors when its basis changes.

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3Blue1Brown: Linear Algebra Chapter 10: Cross Products - Quiz

Cross products and orientation Study the geometric ideas from 3Blue1Brown’s Chapter 10, Cross products, with selected connections to its follow-up lesson on linear transformations and duality. Topics The distinction between a planar signed-area determinant and a three-dimensional vector cross product. Orientation, right-hand conventions, zero products and collinearity. Parallelogram areas, fixed-length angle changes and scaling one or both inputs. Anticommutativity through column swaps and reversal of the normal direction. The symbolic cofactor mnemonic and its connection with the scalar triple product and dot-product duality. Linear maps taking the unit square to a parallelogram. Preparation Review vectors, determinants and basic trigonometry. Distinguish signed quantities from non-negative geometric areas and use an ordered coordinate basis when reasoning about orientation. Some items require selecting several statements. The lesson reference above supplies context for questions about the video’s exposition; the optional follow-up on duality develops the deeper computational connection.

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