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.

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

.

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

.

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.