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- TL;DR Summary
- Change of basis matrix and transformation matrix

Hello,

Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There are infinite possible bases to choose from. Each basis expresses the components as two different numbers.

That said, a particular vector ##X## can be changed (its components can be changed while its direction and magnitude remain the same) from a basis ##A## to a different basis ##B## using a change-of-basis matrix ##M_{AB}##. Vector ##X=(x_1 , x_2)_B##, in basis ##B##, has the same direction and magnitude as vector ##X=(x_1 , x_2)_A## in basis ##A## ,i.e. it remains the same object.

However, there are matrices that can transform a vector ##X## into a different (magnitude and/or direction) vector ##Y## in the same basis ##A##.

What is the difference between a change-of-basis matrix ##M_{AB}## which leaves the vector unchanged and a matrix ##F## that actually changes a vector to a different vector in the same basis ##A##? I know that the columns of matrix ##M## are the basis vector of basis ##A## transformed to basis ##B##. But to accomplish that transformation we need a matrix first.

Matrix ##F##, on the other hand, must change the vector to a different vector but does not affect the basis vectors themselves...

Thank you for any clarification...

Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There are infinite possible bases to choose from. Each basis expresses the components as two different numbers.

That said, a particular vector ##X## can be changed (its components can be changed while its direction and magnitude remain the same) from a basis ##A## to a different basis ##B## using a change-of-basis matrix ##M_{AB}##. Vector ##X=(x_1 , x_2)_B##, in basis ##B##, has the same direction and magnitude as vector ##X=(x_1 , x_2)_A## in basis ##A## ,i.e. it remains the same object.

However, there are matrices that can transform a vector ##X## into a different (magnitude and/or direction) vector ##Y## in the same basis ##A##.

What is the difference between a change-of-basis matrix ##M_{AB}## which leaves the vector unchanged and a matrix ##F## that actually changes a vector to a different vector in the same basis ##A##? I know that the columns of matrix ##M## are the basis vector of basis ##A## transformed to basis ##B##. But to accomplish that transformation we need a matrix first.

Matrix ##F##, on the other hand, must change the vector to a different vector but does not affect the basis vectors themselves...

Thank you for any clarification...