For problem 1-8, verify directly from Definition 6.1.3 that the given mapping is a linear transformation. T : C 2 ( I ) → C 0 ( I ) defined by T ( y ) = ( y ″ + a 1 y ′ + a 2 y ) , where a 1 and a 2 are the functions defined on I .
For problem 1-8, verify directly from Definition 6.1.3 that the given mapping is a linear transformation. T : C 2 ( I ) → C 0 ( I ) defined by T ( y ) = ( y ″ + a 1 y ′ + a 2 y ) , where a 1 and a 2 are the functions defined on I .
Solution Summary: The author explains that the given mapping is a linear transformation. Let V and W be the vector spaces.
. Let f: R²
→ R be defined by f((x, y)) = 9y - 7x. Is f a linear transformation?
a. f((x₁, y₁) + (x2 - y2)) =
f((x₁, y₁>) + f((x₂ - y2)) =
Does f((x₁, y₁) + (x₂ - y₂)) = f({x₁, y₁ )) + f((x₂ - y₂)) for all (x₁, y₁), (x₂, y₂) ER²? choose
b. f(c(x, y)) =
. (Enter x₁ as x1, etc.)
c(f ((x, y))) =
(C
Does f (c(x, y)) = c(f((x, y))) for all c ER and all (x, y) ER²? choose
c. Is f a linear transformation? choose
<
3. Let T R¹ R³ be a linear transformation defined by T(x):
= Ax where the matrix
:
A
-1
-1 -1 0
1
40
0124
2
Chapter 6 Solutions
Differential Equations and Linear Algebra (4th Edition)
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY