Problem 3. (Linear Transformation) Let T: R → R" be a linear transformation with standard matrix A, i.e., for any x € R" we have T(x) = Ax. (1) How many rows and columns are there for the matrix A? Explain your answer. (2) Show that {T(x) : x ≤ R} = {x₁α₁ + ... + £nªn : I1, ..., In R} where a,. an are the column vectors of the matrix A. (3) Show that T is one-to-one if and only if the column vectors of A are linearly independent.

Problem 3. (Linear Transformation) Let T: R → R" be a linear transformation with standard matrix A, i.e., for any x € R" we have T(x) = Ax. (1) How many rows and columns are there for the matrix A? Explain your answer. (2) Show that {T(x) : x ≤ R} = {x₁α₁ + ... + £nªn : I1, ..., In R} where a,. an are the column vectors of the matrix A. (3) Show that T is one-to-one if and only if the column vectors of A are linearly independent.

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter11: Matrices And Determinants

Section11.FOM: Focus On Modeling: Computer Graphics

Problem 6P: Here is a data matrix for a line drawing: D=[012100002440] aDraw the image represented by D. bLet...

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