Movement of a vector when multiplied by a matrix

[Ali Asadullah]

What is the effect on a vector when it is multiplied by a matrix?
Let any matrix
2 3
3 5
What will be the effect on vectors when they are multiplied with this matrix?
In which direction will they move?
What will be the effect of multiplying vectors with any matrix of order n by n?

 

[Mark44]

Different vectors are affected differently when multiplied by a given matrix. If the vector being multiplied by the matrix happens to be an eigenvector, the result vector is a vector in the same (or opposite) direction, scaled by the value of the eigenvalue associated with this eigenvector.

 

[Ali Asadullah]

Let a matrix
1 1
1 1
If we multiply all the points/vectors in a plane with this matrix, what will be the resultant?

 

[berkeman]

Let a matrix
1 1
1 1
If we multiply all the points/vectors in a plane with this matrix, what will be the resultant?

What happens if you try some examples? Can you show us a few examples that you have worked out?

 

[blue_raver22]

When a vector is multiplied by a matrix, the resulting vector will be transformed according to the values in the matrix. In the example given, the matrix has a 2x2 size, which means it can transform a 2-dimensional vector. The effect of this transformation will depend on the specific values in the matrix.

For example, if the vector is [1, 0], the resulting vector after multiplication will be [2, 3]. This means that the vector has moved 2 units in the x-direction and 3 units in the y-direction. Similarly, if the vector is [0, 1], the resulting vector will be [3, 5], meaning a movement of 3 units in the x-direction and 5 units in the y-direction.

In general, multiplying a vector with a matrix of size n by n will result in a transformed vector with n dimensions. The direction and magnitude of the movement will depend on the values in the matrix. This transformation can be used in various applications such as image processing, computer graphics, and data analysis.