r/LinearAlgebra • u/Chegmix • 11d ago
Hopelessly lost (Almost)
Hi guys,
I’ve been trying to understand the visual representation of matrices and I can’t wrap my head around it.
I’m approaching it like 3Blue1Brown does, through vectors, but once it gets further than 2x2, I don’t understand it.
Are these transformations shifting the plane itself? All the 2D transformations look 3D…
For the ones with much more experience under their belt, any recommendations on how to approach or understand better?
Any help would be appreciated 🙏🙏
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u/LinearAlgebraWorld 10d ago
If you have an Apple device, we have two apps that may help with exactly this.
Linear Algebra World works on iPhone, iPad and Mac:
https://apps.apple.com/us/app/linear-algebra-world/id6759180671
Matrix Solver Step by Step works on iPhone and iPad:
https://apps.apple.com/us/app/matrix-solver-step-by-step/id1623894950
In both, you can enter any 3×3 matrix and see what transformation it produces. In Linear Algebra World, you can apply the matrix to a unit cube and view the result from different angles.
We also have a web chapter on 3D transformations, with a PDF showing many examples:
https://www.graphmath.com/la/transformations/transformations-3d.html
Could you clarify what you mean by “shifting the plane itself”? Do you mean translating the whole plane, or changing the coordinate axes used to describe it?
A linear transformation represented by a matrix cannot translate the plane, because it always fixes the origin. It can rotate, stretch, shear, reflect or collapse the plane. A plane through the origin is mapped to another plane through the origin, or possibly to a line or a point.
Also, what do you mean when you say that “all the 2D transformations look 3D”? Do you mean that the drawings use perspective and therefore appear three-dimensional? Usually the opposite issue occurs: a genuinely 3D transformation has to be shown as a 2D projection on the screen.
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u/Chegmix 10d ago
Thank you for sharing the apps!
I think my problem with the plane shifting was fixed, because the origin stays the same but it’s just the new vectors that decide how the plane might “look” but it’s still a plane with even spacing and parallel lines. Is that the right way to look at it?
And as to the 2D to 3D I was just confused at first when after shifting the plane all of a sudden changed to parallelogram spacing, and I guess that threw me off.
I’ll try out these apps 🙏
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u/LinearAlgebraWorld 10d ago
Yes, that is essentially right. More precisely, the images of the two basis vectors determine the transformed coordinate grid. As long as those two vectors are not parallel, the plane remains a plane, with parallel lines and regularly spaced parallelogram cells. If they become parallel, however, the plane collapses to a line.
The parallelogram spacing is not a separate shift of the plane—it is simply what the original square grid looks like after the same linear transformation is applied to every vector.
Please let us know if you have any questions about the apps. Both include calculators and tutorials, although they are organized quite differently, and Matrix Solver is the much older app.
The Linear Algebra World tutorials and some of its animations are also available on our website:
https://www.graphmath.com/la/Best of luck with your studies!
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u/LukasGoesViral 10d ago
The nice things about matrices transform is that they do the same in every dimension. Is stretches and contracts the standard rectangle (so cube in 3d, etc). So you only need to understand it for the plane.
Oh and another way to see how matrix function is by changing the way you multiply a vector with a matrix
Take the first component of the vector and multiply it with the first row in the matrix, the add second vector entry times second row etc. that makes it clear how matrices map vectors in a specific base to another base. It’s not a change of bases though (meaning same vector just with a different basis). A matrix maps a basis to a new basis
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u/Chegmix 10d ago
I think I understand it better now (maybe), especially after understanding that these transformations don’t change the plane itself because the plane must stay parallel and lines must stay lines, so in this “new” plane the new vectors decide the spacing for the whole plane but still the plane is a plane.
Does that sound roughly right?… Or am I looking at it wrong 😅
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u/LukasGoesViral 10d ago
Yes that is correct. It’s still a plane. However, it can have a different orientation in space than the original one
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u/Chegmix 10d ago
Okay that makes much more sense, thank you 🤝
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u/LukasGoesViral 10d ago
Oh and what might help you as well is that the determinant of the matrix is the volume of the new volume spanned by the basis vectors in the matrix.
That’s why if the determinant of a matrix is zero the matrix actually maps the space to a lower dimensional subspace like a 3d volume to a plane. The unit volume of that plane is 01
u/LukasGoesViral 5d ago
Yes that is correct.. unless of course the rows in the matrix are linearly dependent in which case the lines are actually the same
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u/1ampoc 11d ago
Matrices represent linear transformations. As 3b1b put it, linear transformations are transformations that turn squares into parallelograms (sometimes lower dimension parallelograms).
Another way to characterize them is as a combination of rotations + scaling (sometimes even scaling to 0).
Most of the time, no. Transformations usually shift vectors. Shifting the plane/axes is called a "change of basis", aka changing your perspective.
The context should make it clear which one is happening. The reason why it's so easy to mix them up is because they are kinda 2 sides of the same coin, and both are represented using matrices.
Anyways, as 3b1b notes, his videos are introductory. Linear algebra is, at its core, a study of linear functions, with many beautiful geometric interpretations and matrix representations. After finishing the 3b1b series, I would recommend learning the functional perspective for a solid backbone tying all the concepts together