 |
|
|
|
|
|
||||||||||||
|
|||||||||||||||||
 |
 |
|
|
||||||||||||||
 |
 |
 |
|
|
|||||||||||||
 |
|
||||||||||||||||
Matrix Size & Validation
Vector Algebra What is Vector? Vector Norm Unit Vector Vector Addition Vector Subtraction Vector Scalar Multiple Vector Multiplication Vector Inner Product Vector Outer Product Vector Cross Product Vector Triple Cross Product Vector Triple Dot Product Scalar Triple Product Orthogonal & Orthonormal Vector Cos Angle of Vectors Scalar and Vector Projection Matrix Algebra What is a matrix? Special Matrices Matrix One Null Matrix Matrix Diagonal Is Diagonal Matrix? Identity Matrix Matrix Determinant Matrix Sum Matrix Trace Matrix Basic Operation Is Equal Matrix? Matrix Transpose Matrix Addition Matrix Subtraction Matrix Multiplication Matrix Scalar Multiple Hadamard Product Horizontal Concatenation Vertical Concatenation Elementary Row Operations Matrix RREF Finding inverse using RREF (Gauss-Jordan) Finding Matrix Rank using RREF Matrix Inverse Is Singular Matrix? Linear Transformation Matrix Generalized Inverse Solving System of Linear Equations Linear combination, Span & Basis Vector Linearly Dependent & Linearly Independent Change of basis Matrix Rank Matrix Range Matrix Nullity & Null Space Eigen System Matrix Eigen Value & Eigen Vector Symmetric Matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrix Similarity Transformation and Matrix Diagonalization Matrix Power Orthogonal Matrix Spectral Decomposition Singular Value Decomposition Resources on Linear Algebra |
Change Basis In this page you will learn how you can transform a point from one coordinate system into another coordinate system. In the previous topic of Basis Vector, you have learned that we can change the coordinate system. Suppose you have a point with coordinate of
Suppose we start with a point Example:
Example: Note: Transformation of coordinate systems follows equality of matrix-vector multiplication
Rate this tutorial or give your comments about this tutorial Preferable reference for this tutorial is Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\ |
|||||||||||||
|
||||||||||||||
|
||||||||||||||
in Euclidean space (with standard basis coordinate 
and
). We want to find the coordinate of the same point in a new coordinate system form by basis vector 
and
. The figure below shows that the new coordinate of the same point is
. In this page, I will show you how you will obtain the coordinate in the new coordinate system.
in the standard Euclidean basis 
and
. We want to transform it into a new space 
and
. First, we can do 
. Then, the coordinate of a point in the old basis 
is equal to the 
with coordinate of the point in the new basis
. That is
. Thus to get the coordinate of a point in the new basis is the reverse, that is
.
. Our new basis is vector 
and
. Augmenting the basis vectors form a matrix
. The 
. The coordinate of the point in the new coordinate is
. 
from the coordinate system of basis vectors 
and
into Euclidean system. 
. We have our coordinate point 
back.
, where
and
are the matrix of the respective basis vectors. In Euclidean coordinate system, the basis vectors form identity matrix
. Thus, the formula 
can be simplified into 
or
.