Number 23 through 25 linear algebra b. If a finite set S of nonzero vectors spans a vector spaceV, then some subset of S is a basis for V.c. A basis is a linearly independent set that is as large aspossible.d. The standard method for producing a spanning set forNul A, described in Section 4.2, sometimes fails to pro-givenduce a basis for Nul A.e. If B is an echelon form of a matrix A, then the pivotcolumns 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 theset of vectors in R whose second and third entries are equal.Then every vector in H has a unique expansion as a linearcombination of v1, V2, V3, because0.(t -- s)+s 1for any s and t. Is {V1, V2, V3} a basis for H? Why or whynot?26. In the vector space of all real-valued functions, find a basisfor the subspace spanned by {sin t, sin 2t, sin i cos t}.let27. Let V be the vector space of functions that describe thevibration of a mass-spring system. (Refer to Exercise 19 inSection 4.1.) Find a basis for V.S28. (RLC circuit) The circuit in the figure consists of a resistor(R ohms), an inductor (L henrys), a capacitor (C farads), andan 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 avoltmeter.) Let v(t) be the voltage (in volts) at time t,%3D

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Description: Number 23 through 25 linear algebra b. If a finite set S of nonzero vectors spans a vector spaceV, then some subset of S is a basis for V.c. A basis is a linearly independent set that is as large aspossible.d. The standard method for producing a span

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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Number 23 through 25 linear algebra

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

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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