10:24 Done elementary_linear_algebra_8th_edi... O 40. Proof Prove in full detail that M, 2, with the standard operations, is a vector space. 41. Rather than use the standard definitions of addition and scalar multiplication in R², let these two operations be defined as shown below. (a) (x¡, y1) + (x2, y2) c(x, y) (x, + x2, y1 + y2) (сх, у) (b) (x1, y,) + (x2, y,) = (x1, 0) c(x, y) = (cx, cy) (c) (x1, y1) + (x2, y2) = (x1 + xz, Y1 + y2) c(x, y) = (Scx, Scy) With each of these new definitions, is R² a vector space? Justify your answers. 42. Rather than use the standard definitions of addition and scalar multiplication in R³, let these two operations be defined as shown below. (a) (x1, y1, z,) + (x2, V2, Z2) (x, + x2, y1 + y2, Z1 + z2) с(х, у, г) %3 (сх, су, 0) (b) (x1, y1, z1) + (x2, Y2, 72) = (0, 0, 0) с(х, у, г) — (сх, су, сг) (c) (x1, y1, Z1) + (x,, Y2, Z2) (x1 + x, + 1, y, + y2 + 1, z1 + z2 + 1) с (х, у, г) — (сх, су, сг) (d) (x1, y1, Z1) + (x2, Y2, 72) %3D (x, + x, + 1, y, + y2 + 1, z, + z2 + 1) 1, cy + c – 1, cz + c – 1) с(х, у, г) 3 (сх + с With each of these new definitions, is R³ a vector space? Justify your answers. 43. Prove that in a given vector space V, the zero vector is unique. 44. Prove that in a given vector space V, the additive inverse of a vector is unique. 45. Mass-Spring System The mass in a mass-spring system (see figure) is pulled downward and then released, causing the system to oscillate according to x(t) = a, sin w wy vo wr
10:24 Done elementary_linear_algebra_8th_edi... O 40. Proof Prove in full detail that M, 2, with the standard operations, is a vector space. 41. Rather than use the standard definitions of addition and scalar multiplication in R², let these two operations be defined as shown below. (a) (x¡, y1) + (x2, y2) c(x, y) (x, + x2, y1 + y2) (сх, у) (b) (x1, y,) + (x2, y,) = (x1, 0) c(x, y) = (cx, cy) (c) (x1, y1) + (x2, y2) = (x1 + xz, Y1 + y2) c(x, y) = (Scx, Scy) With each of these new definitions, is R² a vector space? Justify your answers. 42. Rather than use the standard definitions of addition and scalar multiplication in R³, let these two operations be defined as shown below. (a) (x1, y1, z,) + (x2, V2, Z2) (x, + x2, y1 + y2, Z1 + z2) с(х, у, г) %3 (сх, су, 0) (b) (x1, y1, z1) + (x2, Y2, 72) = (0, 0, 0) с(х, у, г) — (сх, су, сг) (c) (x1, y1, Z1) + (x,, Y2, Z2) (x1 + x, + 1, y, + y2 + 1, z1 + z2 + 1) с (х, у, г) — (сх, су, сг) (d) (x1, y1, Z1) + (x2, Y2, 72) %3D (x, + x, + 1, y, + y2 + 1, z, + z2 + 1) 1, cy + c – 1, cz + c – 1) с(х, у, г) 3 (сх + с With each of these new definitions, is R³ a vector space? Justify your answers. 43. Prove that in a given vector space V, the zero vector is unique. 44. Prove that in a given vector space V, the additive inverse of a vector is unique. 45. Mass-Spring System The mass in a mass-spring system (see figure) is pulled downward and then released, causing the system to oscillate according to x(t) = a, sin w wy vo wr
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Chapter1: Functions And Models
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Done elementary_linear_algebra_8th_edi... O
40. Proof Prove in full detail that M, 2, with the standard
operations, is a vector space.
41. Rather than use the standard definitions of addition
and scalar multiplication in R², let these two operations
be defined as shown below.
(a) (x¡, y1) + (x2, y2)
c(x, y)
(x, + x2, y1 + y2)
(сх, у)
(b) (x1, y,) + (x2, y,) = (x1, 0)
c(x, y) = (cx, cy)
(c) (x1, y1) + (x2, y2) = (x1 + xz, Y1 + y2)
c(x, y) = (Scx, Scy)
With each of these new definitions, is R² a vector space?
Justify your answers.
42. Rather than use the standard definitions of addition and
scalar multiplication in R³, let these two operations be
defined as shown below.
(a) (x1, y1, z,) + (x2, V2, Z2)
(x, + x2, y1 + y2, Z1 + z2)
с(х, у, г) %3 (сх, су, 0)
(b) (x1, y1, z1) + (x2, Y2, 72) = (0, 0, 0)
с(х, у, г) — (сх, су, сг)
(c) (x1, y1, Z1) + (x,, Y2, Z2)
(x1 + x, + 1, y, + y2 + 1, z1 + z2 + 1)
с (х, у, г) — (сх, су, сг)
(d) (x1, y1, Z1) + (x2, Y2, 72)
%3D
(x, + x, + 1, y, + y2 + 1, z, + z2 + 1)
1, cy + c – 1, cz + c – 1)
с(х, у, г) 3 (сх + с
With each of these new definitions, is R³ a vector space?
Justify your answers.
43. Prove that in a given vector space V, the zero vector is
unique.
44. Prove that in a given vector space V, the additive
inverse of a vector is unique.
45. Mass-Spring System The mass in a mass-spring
system (see figure) is pulled downward and then
released, causing the system to oscillate according to
x(t) = a, sin w
wy vo wr
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