Sets of Linear Combinations and Their Images under Linear Transformations

This Demonstration visualizes points, vectors, and the effect of linear transformations from to . The black and red vectors are linear combinations of and given by and , respectively. The green dots are integral scalar multiples of , and , and the black dots show the images of the green dots under the linear transformation defined by the matrix .


  • [Snapshot]
  • [Snapshot]
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This Demonstration can be used to support linear algebra activities on vector space concepts: geometric vector, linear combination, linear independence, spanning set, span, basis, coordinates, and so on; and topics on linear transformations: domain, range, kernel (null space), orthogonal projection, one-to-one and onto transformations. For further explanations and the definition of the controls, see below the descriptions of the three Snapshots.
Snapshot 1
The blue line represents the vector whose component values are assigned using the symbols , , and .
The purple line represents the vector whose component values are assigned using the symbols , , and .
The light blue line represents the vector whose component values are assigned using the symbols , , and .
The red dot represents the origin, .
The brown dot represents a point whose component values are assigned using the symbols , , and .
Snapshot 2
The green points are the linear combinations of the vectors , , and for scalars , , and obtained by setting the sliders , and and using values for scalars ranging from to .
The black vector, u, corresponds to the linear combination of the vectors and .
The red vector, v, corresponds to the linear combination of the vectors , , and .
Snapshot 3
The black points are the images of linear transformations of the green points. The symbol stands for 3×3 matrices representing linear transformations, and their matrix entries are obtained by assigning values to the symbols , .
One such linear transformation is represented by the matrix , which maps vectors along the axis to vectors along the axis (that is, maps the greeen points on the axis to black points on the axis). The transformation preserves magnitutes and maps vectors along the axis to themselves.
Under this transformation, any vector (linear combinations of vectors and ) on the green plane are mapped to the vectors on the black plane. For example, the image of the black vector, , (obtained from the linear combination ) on the green plane can easily be located on the black plane using the linear combination . Once the coordinates of the image of the black vector, , are identified, this image can furthermore be verified by the brown point, , as seen in Snapshot 3. Here, you may observe that the location of the point traces out the same path as the black vector. This time, the tracing of three points is done along the axis rather than along the vector , keeping the tracing of two points the same, parallel to vector on the axis.
[1] H. Dogan-Dunlap, "Linear Algebra Students’ Modes of Reasoning: Geometric Representations," Linear Algebra and Its Applications 432, pp. 2141–2159.
[2] H. Dogan, R. Carrizales, and P. Beaven, Metonymy and Object Formation: Vector Space Theory. In Ubuz, B. (Ed.) Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, (Research Reports) Vol. 2, 2011, pp. 265–272. Ankara, Turkey: PME.
[3] L. W. Johnson, R. D. Riess, and J. T. Arnold, Introduction to Linear Algebra, 5th ed., Pearson Education, 2002.
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