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sportlover36
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in my book it asks me to give an example of a linear transformation T: p4 -> R^4 that's onto. I have to prove that T is onto and a linear transformation...can someone give me some advice?
A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the operations of vector addition and scalar multiplication. In other words, it takes in a vector and outputs another vector that is still within the same vector space.
A linear transformation is onto if every element in the output space is mapped to by at least one element in the input space. In other words, every vector in the output space has a pre-image in the input space.
Yes, the transformation T: R^3 -> R^2 defined by T(x, y, z) = (x + y, 2y + z) is onto. This means that for any vector (a, b) in R^2, there exists a vector (x, y, z) in R^3 such that T(x, y, z) = (a, b).
To determine if a linear transformation is onto, you can use the rank-nullity theorem, which states that the rank of the transformation plus the dimension of the null space must equal the dimension of the input space. If the rank of the transformation is equal to the dimension of the output space, then the transformation is onto.
A linear transformation being onto means that every element in the output space has a pre-image in the input space, which is important for applications in mathematics, physics, and engineering. It allows for the manipulation and transformation of data in a way that preserves important properties and relationships, making it a useful tool in problem solving and analysis.