Linear transformation of matrices are linear operations using matrices that modify the initial dimension of a given vector.
In other words, we can modify the dimension of a vector by multiplying it by any matrix.
Linear transformations are the basis of the vectors and eigenvalues of a matrix since they depend linearly on each other.
Recommended articles: operations with matrices , vectors and eigenvalues .
We define any matrix C of dimension 3 × 2 multiplied by a vector V of dimension n = 2 such that V = (v 1 , v 2 ).
What dimension will the result vector be?
The vector resulting from the product of the matrix C 3 × 2 with the vector V 2 × 1 will be a new vector V ‘of dimension 3.
This change in dimension of the vector is due to the linear transformation by the matrix C .
Given the square matrix R with dimension 2 × 2 and the vector V of dimension 2.
A linear transformation of the dimension of vector V is:
where the initial dimension of vector V was 2 × 1 and now the final dimension of vector V is 3 × 1. This change in dimension is achieved by matrix multiplication R .
Can these linear transformations be represented graphically? Well of course!
We will represent the result vector V ‘on a plane.
V = (2.1)
V ‘= (6.4)
Eigenvectors by graphical representation
How can we determine that a vector is an eigenvector of a given matrix just by looking at the graph?
We define the 2 × 2 dimension matrix D :
Are the vectors v 1 = (1,0) and v 2 = (2,4) eigenvectors of the matrix D ?
1. Let’s start with the first vector v 1 . We do the above linear transformation:
So if vector v 1 is actually eigenvector of matrix D , the resulting vector v 1 ‘and vector v 1 should belong to the same line.
We represent v 1 = (1,0) and v 1 ‘= (3,0).
Since both v 1 and v 1 ‘belong to the same line, v 1 is an eigenvector of the matrix D .
Mathematically, there is a constant h (eigenvalue) such that:
2. We continue with the second vector v 2 . We repeat the above linear transformation:
So, if vector v 2 is really eigenvector of matrix D , the resulting vector v 2 ‘and vector v 2 should belong to the same line (like the previous graph).
We represent v 2 = (2,4) and v 2 ‘= (2,24).
Since v 2 v 2 ‘non Similarly, v 2 is an eigenvector of the matrix D .
Mathematically, there is no constant h (eigenvalue) such that: