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esler21
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Is L:R^2 - ->R^2 defined by L(x,y) = (x-1,y-x) a linear transformation? Explain why or why not.
A linear transformation is a function or mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, for a linear transformation to exist, it must satisfy the properties of additivity and homogeneity.
To determine if L:R^2 -> R^2 is a linear transformation, you can check if it satisfies the properties of additivity and homogeneity. This means that for any two vectors u and v in R^2 and any scalar c, the following must hold true:
L(u + v) = L(u) + L(v)
L(cu) = cL(u)
A linear transformation is onto if every vector in the target space R^2 can be mapped to by at least one vector in the domain R^2. In other words, the range of the linear transformation covers the entire target space.
While the terms "linear transformation" and "linear function" are often used interchangeably, there is a small distinction between the two. A linear transformation is a mapping between vector spaces, while a linear function is a mapping between real numbers.
No, by definition, a linear transformation must preserve the properties of additivity and homogeneity. If these properties are not satisfied, the transformation is not considered linear.