Suppose R* = is a basis for R*. Span {v1,..., V4}. Explain why {V1,...,V4} %3D
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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Number 23 through 25 linear algebra

Transcribed Image Text:b. If a finite set S of nonzero vectors spans a vector space
V, then some subset of S is a basis for V.
c. A basis is a linearly independent set that is as large as
possible.
d. The standard method for producing a spanning set for
Nul A, described in Section 4.2, sometimes fails to pro-
given
duce a basis for Nul A.
e. If B is an echelon form of a matrix A, then the pivot
columns of B form a basis for Col A.
23. Suppose R“ =
is a basis for R*.
Span {v1,..., vA}. Explain why {v1,..., V4}
24. Let B =
{V1, ..,Vn} be a linearly independent set in R".
Explain why B must be a basis for R".
25. Let vi =
V2 =
V3 =
and let H be the
set of vectors in R whose second and third entries are equal.
Then every vector in H has a unique expansion as a linear
combination of v1, V2, V3, because
0.
(t -- s)
+s 1
for any s and t. Is {V1, V2, V3} a basis for H? Why or why
not?
26. In the vector space of all real-valued functions, find a basis
for the subspace spanned by {sin t, sin 2t, sin i cos t}.
let
27. Let V be the vector space of functions that describe the
vibration of a mass-spring system. (Refer to Exercise 19 in
Section 4.1.) Find a basis for V.
S
28. (RLC circuit) The circuit in the figure consists of a resistor
(R ohms), an inductor (L henrys), a capacitor (C farads), and
an initial voltage source. Let b = R/(2L), and suppose R,
L, and C have been selected so that b also equals 1/LC.
(This is done, for instance, when the circuit is used in a
voltmeter.) Let v(t) be the voltage (in volts) at time t,
%3D
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