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They applied the operator T to the vector using either the explicit form of the matrix T just above or by using the original definition.karush said:how did they get these (red)
"Find [T]ββ for NMH{823}" is asking for the transformation matrix [T] from the standard basis β to the basis β for the vector space NMH{823}. This matrix represents how the coordinates of a vector in the standard basis would change when expressed in the basis β.
To find the transformation matrix [T]ββ, you will need to know the basis vectors for the standard basis β and the basis β for NMH{823}. Then, you can use the formula [T]ββ = [I]ββ^-1, where [I]ββ is the identity matrix with the basis vectors for β as its columns, and ^-1 represents the inverse operation.
The purpose of finding [T]ββ is to be able to convert coordinates of vectors between different bases. This is useful in various applications, such as solving systems of linear equations, performing transformations in geometry, and working with different coordinate systems in physics and engineering.
Yes, the transformation matrix [T]ββ can be different for different vector spaces. This is because the basis vectors for each vector space can be different, and the transformation matrix is dependent on the basis vectors.
There are some shortcuts or tricks that can be used to find [T]ββ, such as using row reduction techniques or using the change of basis formula. However, it is important to understand the concept and formula for [T]ββ in order to use these shortcuts effectively.