Nexttime35 said:Homework Statement
Let A = [1 0
4 2 ]
Let B be the eigenbasis {[1,4], [0,1]}.
--Find [T]B where T(x)=A(x).The Attempt at a Solution
Would [T]B = {[1,-1], [0,2]}?
We are trying to find [T]B, the matrix representation of T with respect to B. So would my answer be correct?
Thanks.
Zondrina said:If I'm reading correctly, you want to find ##[T]_B##. This amounts to finding the image of the basis vectors under ##T##.
Nexttime35 said:Yes, I want to find ##[T]_B## . I guess I am confused about how to find the basis for im(T). Could you possibly point me in the right direction?
Thank you for your help.
The "Change of Basis Problem" is a mathematical concept that involves representing a vector or a linear transformation in terms of a new basis. It is often used in linear algebra to simplify calculations and solve problems in different coordinate systems.
The "Change of Basis Problem" is important because it allows us to work with vectors and linear transformations in different coordinate systems. This can be helpful in solving various problems in physics, engineering, and other fields where different coordinate systems are used.
The "Change of Basis Problem" has many real-world applications, including computer graphics and computer vision, where it is used to rotate and translate objects in 3D space. It is also used in GPS navigation systems, where it is used to convert coordinates between different map projections.
To change the basis of a vector, we need to find a transformation matrix that maps the original basis to the new basis. This transformation matrix is usually obtained by writing the basis vectors of the new basis as a linear combination of the basis vectors of the original basis. Multiplying the vector by this transformation matrix gives us the coordinates of the vector in the new basis.
Yes, the "Change of Basis Problem" is unique. This means that for any given vector or linear transformation, there is only one way to represent it in terms of a new basis. However, the transformation matrix may vary depending on the choice of the new basis.