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Problem 1. Consider the vectors 3 Let {e1, e2, es} be the standard basis in R° and let T : R³ → R* be the linear transformation defined by T(e1) = V1, T(e2) = v2, T(e3) = V3. (a) Write down the matrir Ar such that T(x) = ATX. (b) What is the dimension of R(T), the range of T? Is the vector w in R(T')? Justify your answers. (c) State the rank-nullity theorem for linear transformations and use it to determine the dimension of the null space N(T) of T.

Problem 1. Consider the vectors 3 Let {e1, e2, es} be the standard basis in R° and let T : R³ → R* be the linear transformation defined by T(e1) = V1, T(e2) = v2, T(e3) = V3. (a) Write down the matrir Ar such that T(x) = ATX. (b) What is the dimension of R(T), the range of T? Is the vector w in R(T')? Justify your answers. (c) State the rank-nullity theorem for linear transformations and use it to determine the dimension of the null space N(T) of T.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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aS FAST as possible please! please provide a typewritten solution i would be grateful! PROBLEM 1 

Problem 1. Consider the vectors (0-0-0-0 Let {e,, e2, es} be the standard basis in R3 and let T : R³ → R' be the linear transformation defined by T(e1) = V1, T(e2) = v2, T(e3) = V3. (а) Write down the matrir Ar such that T(x) — Арх. (b) What is the dimension of R(T), the range of T? Is the vector w in R(T)? Justify your answers. (c) State the rank-nullity theorem for linear transformations and use it to determine the dimension of the null space N(T) of T. (d) Find a basis of N(T).
Transcribed Image Text:Problem 1. Consider the vectors (0-0-0-0 Let {e,, e2, es} be the standard basis in R3 and let T : R³ → R' be the linear transformation defined by T(e1) = V1, T(e2) = v2, T(e3) = V3. (а) Write down the matrir Ar such that T(x) — Арх. (b) What is the dimension of R(T), the range of T? Is the vector w in R(T)? Justify your answers. (c) State the rank-nullity theorem for linear transformations and use it to determine the dimension of the null space N(T) of T. (d) Find a basis of N(T).
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