Change of Basis Matrix - Transition Matrix
- change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis
- change of basis matrix is the matrix whose columns are the coordinate vectors of the new basis vectors on the old basis
Change of Basis - Intuition
Understanding how a linear transformation 𝑇 transforms standard basis vectors 𝑖 and 𝑗.
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For example, given the following linear transformation 𝑇:
and also the following basis vectors 𝑖 and 𝑗 (in this example 𝑖 and 𝑗 are the standard basis vectors):
Taking the transformation of 𝑇 on 𝑖 and 𝑗:
This is what it looks like graphically
Understanding how a linear transformation 𝑇 transforms an arbitrary vector 𝑣, where 𝑣 is expressed by basis vectors 𝑖 and 𝑗
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For example, given the following linear transformation 𝑇, basis vectors 𝑖 and 𝑗, and vector 𝑣:
Vector 𝑣 can be decomposed into its constituent basis vectors:
Taking the transformation of 𝑇 on 𝑣 is the same as taking the transformations of the basis vectors 𝑖 and 𝑗 individually:
This is what it looks like geometrically
Change of Basis Matrix - Intuition
A Vector Expressed in a Different Basis
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Say we have the following basis vectors 𝑖 and 𝑗 defined as:
Given a vector 𝑣 expressed under those basis vectors, what would that vector be when expressed under the standard basis vectors?
First, take the basis vectors 𝑖 and 𝑗 as columns to form matrix 𝐶 like so:
Then apply the transformation 𝐶 onto vector 𝑣 like so:
Indent
𝐶𝑣
Given a vector expressed under the standard basis vectors, what would that vector be when expressed under the basis vectors 𝑖 and 𝑗?
-1𝑣
𝐶
A Matrix Transformation Expressed in a Different Basis
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Say we have a transformation matrix 𝑇 expressed under the standard basis vectors. How would we describe 𝑇 in a different set of basis vectors 𝑖 and 𝑗?
Say the following basis vectors 𝑖 and 𝑗 are defined as:
Thus, the change of basis matrix 𝐶 is defined as:
The transformation matrix 𝑇 expressed in basis 𝑖 and 𝑗 is defined as:
-1𝑇𝐶
𝐶
𝐶-1𝑇𝐶 can be thought of as:
- 𝐶 transforms a vector expressed from basis 𝑖 and 𝑗 to the standard basis
- 𝑇 transforms a vector expressed in the standard basis
- 𝐶-1 transforms a vector expressed from the standard basis to basis 𝑖 and 𝑗