4. Let f1, f2, fz be functions in F(R). a. For a set of real numbers x1, X2, X3, let (F(x;)) be the 3-by-3 matrix whose (i, j) entry is f(x;), for 1< i, j s 3. Prove that the functions f1, f2, f3 are linearly independent if the rows of the matrix (f(x})) are linearly independent. b. Assume the functions f1, f2, f3 have first and second derivatives on some interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is fY-", for 1 < i, j< 3, where f'0) = f, f(1) = f' and f(2) = f" for a dif- ferentiable function f. Prove that f1, f2, f3 are linearly independent if for some x in (a, b), the rows of the matrix W(x) are linearly independent. Show that the following sets of functions are linearly independent. %3D c. f;(x) = –x² + x + 1, f2(x) = x² + 2x, f3(x) = x² – 1. d. fi(x) = e¯*, f2(x) = x, f3(x) = e²ª. e. f,(x) = e*, f2(x) = sin x, f3(x) Note that if the tests in (a) or (b) fail, it is not guaranteed that the functions are linearly dependent. %3D = cos x.
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
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Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
(b) only
![4. Let f1, f2, f3 be functions in F(R).
a. For a set of real numbers x, X2, X3, let (f(x,)) be the 3-by-3 matrix
whose (i, j) entry is f(x;), for 1 < i, js 3. Prove that the functions
f1, f2, f3 are linearly independent if the rows of the matrix (f(x;)) are
linearly independent.
b. Assume the functions f1, f2, f3 have first and second derivatives on some
interval (a, b), and let W(x) be the 3-by-3 matrix whose (i, j) entry is
fU-), for 1 < i, j< 3, where f(0) = f, f(1) = f' and f(2) = f" for a dif-
ferentiable function f. Prove that f, f2, f3 are linearly independent if
for some x in (a, b), the rows of the matrix W(x) are linearly independent.
Show that the following sets of functions are linearly independent.
c. f;(x) = -x² + x + 1, f2(x) = x² + 2x, f3(x) = x² – 1.
d. f,(x) = e¯*, f2(x) = x, f,(x) = e2ª.
e. fi(x) = e*, f2(x) = sin x, f3(x) = cos x.
Note that if the tests in (a) or (b) fail, it is not guaranteed that the functions are
linearly dependent.
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6915687c-4f06-4661-ac02-aefc88931b4a%2F300377d1-0af4-400a-9559-4f8152f9edb6%2Fqlhb73n_processed.png&w=3840&q=75)
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