synapticz

3.2 Building a Matrix from Basis Images

by Synapticz Media
Oct 20, 2026
9 quizzes
Public
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Module 3: Linear Transformations and Matrix Algebra. Building a Matrix from Basis Images. Microlearning practice develops conceptual reasoning, geometric interpretation and carefully checked calculation.

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9 quizzes

Showing 1–9 of 9 quizzes · Page 1 of 1

3.2-01 Why basis images determine a map

3.2-01 Why basis images determine a map

Why basis images determine a map — linear algebra practice Connect a vector’s basis expansion to the corresponding combination of its images. Work with standard and nonstandard input bases, recover unknown basis images from sums, and distinguish complete information from partial or inconsistent data. The linearity assumption remains explicit throughout. What you will practise Use basis coefficients to predict outputs Recover images from combined inputs Check consistency and uniqueness of defining data This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review basis expansions and linearity. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to build a standard matrix correctly.

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3.2-02 Constructing the standard matrix

3.2-02 Constructing the standard matrix

Constructing the standard matrix — linear algebra practice Translate basis images and coordinate formulas into standard matrices. Emphasis falls on column placement, rectangular dimensions, missing entries and verification at new inputs. Nonstandard input data and an extra consistency check extend the routine construction into a useful method for recovering an unknown map. What you will practise Place images in columns Handle rectangular maps Recover and verify matrices from supplied input-output data This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review standard basis vectors and matrix-vector products. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to represent geometric transformations algebraically.

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3.2-03 Scaling and contraction

3.2-03 Scaling and contraction

Scaling and contraction — linear algebra practice Investigate diagonal maps that stretch, contract, reverse or erase coordinate directions. Compare uniform scaling with axis-dependent behavior, compute transformed vectors, recover scale factors and analyze which vectors keep their length or remain fixed. Zero and negative scale factors are treated explicitly. What you will practise Build diagonal scaling matrices Calculate and recover axis factors Identify fixed vectors and lost directions This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review coordinates, lengths and standard matrices. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to separate scaling effects from rotations and shears.

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3.2-04 Reflections

3.2-04 Reflections

Reflections — linear algebra practice Construct reflections across coordinate axes, both diagonal lines and a general line through the origin. Use parallel and perpendicular components to justify matrices and check outputs. The distinction between reflection through an origin-passing line and an offset affine reflection is part of the progression. What you will practise Track fixed and reversed directions Derive arbitrary-line reflections Recognize reflection matrices and affine exceptions This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review basis images, dot products and projection onto a direction. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to combine reflections with other geometric maps.

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3.2-05 Two-dimensional rotations

3.2-05 Two-dimensional rotations

Two-dimensional rotations — linear algebra practice Derive rotation matrices from the two standard basis images under a fixed angle. Calculate clockwise and counterclockwise turns, use exact trigonometric values, recover a matrix from a known basis image and interpret inverse and transpose relationships. Angle conventions and input-vector linearity are kept distinct. What you will practise Construct fixed-angle rotation matrices Calculate exact rotated vectors Explain invariants and inverse-angle behavior This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review plane angles, sine, cosine and basis images. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to compose planar transformations with reliable signs.

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3.2-06 Three-dimensional axis rotations

3.2-06 Three-dimensional axis rotations

Three-dimensional axis rotations — linear algebra practice Use right-handed axes, active rotations and column vectors to construct rotations about coordinate axes. Follow basis directions and concrete points, identify the fixed axis and compare successive turns about different axes. A final convention check explains why apparently conflicting formulas need careful interpretation. What you will practise Apply the right-hand convention Construct axis rotation matrices Track fixed directions and ordered compositions This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review planar rotations and three-coordinate vectors. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to work with spatial graphics transforms.

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3.2-07 Shears

3.2-07 Shears

Shears — linear algebra practice Build horizontal and vertical shear matrices by following basis directions. Calculate shifts, recover a shear parameter and inspect the image of a square. Fixed axes, changing lengths and angles, reversal of a shear, and repeated shears provide geometric meaning beyond entry-by-entry calculation. What you will practise Distinguish horizontal and vertical shears Recover parameters from point images Analyze fixed directions and compositions This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review standard matrices and plane coordinates. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to compare transformation order geometrically.

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3.2-08 Projections and collapse

3.2-08 Projections and collapse

Projections and collapse — linear algebra practice Construct coordinate and line projections, compute their outputs and identify erased directions. Compare orthogonal projection with an oblique projection, explain repeated projection, and examine a plane specified by its normal. The exercises connect the retained component with the information that cannot be recovered. What you will practise Compute line and plane projections Identify image and erased directions Distinguish orthogonal and oblique projection This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review dot products, coordinate subspaces and basis images. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to reason about information loss and injectivity.

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3.2-09 Recovering a matrix from a picture

3.2-09 Recovering a matrix from a picture

Recovering a matrix from a picture — linear algebra practice Read labeled coordinate diagrams and convert transformed basis arrows into matrix columns. The included text graphics state their unit scales and endpoint labels. Later tasks address nonunit input arrows, shifted arrow tails, collapsed grids and inconsistent diagrams, highlighting the information a drawing must supply. What you will practise Read basis-image coordinates from diagrams Account for scale and arrow displacement Check a diagram against linearity This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies. Before starting, review coordinate grids and standard matrix construction. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to recover transformations from visual data.

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