d) Let A be a m × n matrix with real number entries. By considering the linear map TA : R" → R" given by TA(x) = Ax, prove that i) if n < m then there is no matrix B such that AB = Im, and ii) if m < n then there is no matrix B such that BA = I,. 2 1 e) Let A = -1 -4 -2 1. Find a matrix B whose entries are real numbers such that AB = I3, justifying your answer. -1 -9 -2

d) Let A be a m × n matrix with real number entries. By considering the linear map TA : R" → R" given by TA(x) = Ax, prove that i) if n < m then there is no matrix B such that AB = Im, and ii) if m < n then there is no matrix B such that BA = I,. 2 1 e) Let A = -1 -4 -2 1. Find a matrix B whose entries are real numbers such that AB = I3, justifying your answer. -1 -9 -2

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

d) Let A be a m × n matrix with real number entries. By considering the linear map TA : R" → R" given by TA(x) = Ax, prove that
i) if n < m then there is no matrix B such that AB = Im, and
ii) if m < n then there is no matrix B such that BA
In.
2
1
e) Let A
-4
-2
Find a matrix B whose entries are real numbers such that AB = I3, justifying your answer.
-9
-2 2

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The cross product of two vectors in R³ is defined by Let ✓ = A = [1] 8 . Find the matrix A of the linear transformation from R³ to R³ given by T(x) = × đ. a2 X b₁ b₂ b3 = [a2b3-a3b₂] | a3b1 — a1b3 [a1b₂-a2b₁]
Let P0, P1 and P2 be points in R2.