Linear Algebra
1. NulA, ColA
2. Reflection over a plane in 3D.
3. Orthogonal Projection on a plane in 3D.
4. Orthogonal Projection on a line.
5. Matrices multiplication
6. Determinant
7. Vector in 3D.
8. Linear combination
9. Sum of two vectors.
10. Scalar multiplication
11. Linear transformation from 2D space to 3D space.
12. Linear transformation from 3D space to 2D space.
13. Matrix transformation
14. Vector equation.
15. Eigenvectors and Eigenvalues in 3D
16. Transformation matrices
17. Matrices Used To Generate Fibonacci Sequence
18. Gershgorin circle theorem
19. Symmetric Matrix Characterization
20. Exploring Eigenvectors and Eigenvalues Visually
21. Perform row operations on a matrix
22. Roots of a complex number
23. Find the area of a triangle using a determinant.

Linear Algebra

YoavDvir

1. NulA, ColA
2. Reflection over a plane in 3D.
3. Orthogonal Projection on a plane in 3D.
4. Orthogonal Projection on a line.
5. Matrices multiplication
6. Determinant
7. Vector in 3D.
8. Linear combination
9. Sum of two vectors.
10. Scalar multiplication
11. Linear transformation from 2D space to 3D space.
12. Linear transformation from 3D space to 2D space.
13. Matrix transformation
14. Vector equation.
15. Eigenvectors and Eigenvalues in 3D
16. Transformation matrices
17. Matrices Used To Generate Fibonacci Sequence
18. Gershgorin circle theorem
19. Symmetric Matrix Characterization
20. Exploring Eigenvectors and Eigenvalues Visually
21. Perform row operations on a matrix
22. Roots of a complex number
23. Find the area of a triangle using a determinant.

NulA, ColA

Reflection over a plane in 3D.

Orthogonal Projection on a plane in 3D.

Orthogonal Projection on a line.

4.

To find the Orthogonal Projection of

$Y$ on the line of

$U$ ,
Drag

$\alpha U$ on the Line of

$U$ until

$Y-\alpha U$ is perpendicular to

$U$ .

Matrices multiplication

5.

Composition of linear transformations.
Set matrices A,B and then set vector X.

[What should be the value of

$X$ so that

$Z$ would be equal to

$c_1$ ?

Determinant

6.

Set tha values of matrix A. Then drag the verteces of the left polygon. See how the areas of the two polygons change, while the ratio stays fix and depends only on the matrix values.

Vector in 3D.

Linear combination

Sum of two vectors.

Scalar multiplication

Linear transformation from 2D space to 3D space.

11.

Set matrix A and vector X on the spreadsheet.

Linear transformation from 3D space to 2D space.

12.

Set matrix

$A$ on the spreadsheet and then set vector

$X$ .

Matrix transformation

Vector equation.

14.

Find a geometric solution for the following vector equation.

Eigenvectors and Eigenvalues in 3D

15.

Set

$d_1, d_2, d_3$ and drag

$X_1, X_2, X_3$ on the right board.

Transformation matrices

Moving the blue points on the left will change the transformation matrix. The transformation you define is then applied to the quadrilateral on the right hand side. You can move this quadrilateral around to see the effect of the transformation.

Matrices Used To Generate Fibonacci Sequence

17.

A vector containing the two initial term values of a Fibonacci Sequence is repeatedly multiplied by a 2 X 2 matrix, producing subsequent terms of the sequence.

Gershgorin circle theorem

18.

Symmetric Matrix Characterization

19.

Adjust sliders a, b, and c to change the matrix M. This changes the function F=xT*M*x so the constant surfaces of F (ellipses or hyperbolas) are modified in the view.
Move the blue point at the tip of the red arrow to change the x vector of the equation Mx=b. The blue vector b is the result of the transform. The dashed blue vector, is the vector b moved to the tip of the vector x. Note that this vector is normal to the constant surfaces of the function F, and is actually equivalent to half the gradient of the function F.

The purple arrow is resticted to the constant surface F=1. Move the green point at the tip of the purple arrow to change the position of this vector. The green arrow is the result of M times the purple arrow. A right triangle symbol is shown in order to indicate that the resulting green vector is always normal to the constant surface F=1 since it is 1/2 the gradient of F.

Exploring Eigenvectors and Eigenvalues Visually

20.

Drag the point "Drag me" at the tip of the vector u until the vectors u and M u are parallel. At this point, the relation M u = lambda; u is satisfied and u is an eigenvector and λ is an eigenvalue.

If M u and u point in the same direction, then lambda; is positive. If M u and u point in opposite directions, then lambda; is negative.

You can introduce a new matrix by editing M in the Algebra View.

Perform row operations on a matrix

21.

Roots of a complex number

22.

Z and W are complex numbers.
Find the

$n^{th}$ roots of Z by dragging W
such that

$W^n$ meets Z.
How many roots has the

$n^{th}$ of Z, and why?

Find the area of a triangle using a determinant.

23.

Grab a vertex and move it or move the sliders to change the coordinates of the vertices of the triangle.
The area is given by 1/2 of the absolute value of the determinant.
See if you can move the vertices so the value of the determinant becomes negative.