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true or false: if a is an m x n matrix and t is a transformation for which t(x) = ax, then the range of the transformation is t is r^m

User Trillion
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False.

The range of the transformation T is not necessarily equal to R^m.

The range of a linear transformation T: R^n -> R^m is the set of all possible output vectors of T, i.e., the set of all vectors y in R^m such that there exists an input vector x in R^n such that T(x) = y.

The range of a transformation T can be thought of as the span of the columns of the matrix A that represents T, which is the set of all possible linear combinations of the columns of A.

Therefore, the range of the transformation T will depend on the column space of A, which is a subspace of R^m, and not necessarily equal to R^m. The dimension of the column space of A will give the rank of the matrix A, and the rank of A can be at most min(m, n).

User Mahamudul Hasan
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