Problem 6.1. In each of the following, a subset W of some n-space R" is de- fined by means of a given condition imposed on the typical vector (₁,...,In). Determine whether W is a subspace of R". (a) W is the set of all vectors in R³ such that 23 = 0. (b) W is the set of all vectors in R³ such that x₁ = 5x2. (c) W is the set of all vectors in R³ such that x₂ = 1. (d) W is the set of all vectors in R³ such that x₁ + x₂ + x3 = 1. (e) W is the set of all vec- in R4 such tors that x1 + 2x2 + 3x3 + 3x4 = 0. (f) W is the set of all vectors in R4 such that 1 3x3 and x₂ = 4x4. (g) W is the set of all vectors in R2 such that r₁| = |x₂|. (h) W is the set of all vectors in R2 such that (1₁)² + (x₂)² = 0. (i) W is the set of all vectors in R2 such that (x₁)² + (x₂)² = 1. (j) W is the set R2 such that of all vectors in ₁| + |₂| = 1. (k) W is the set of all vectors in R such that x₁ + x₂ = x3 + x4. (1) W is the set of all vectors in R4 such that x₁x2 = x3x4- (m) W is the set of all vectors in R4 such that x1x2x3x4 = 0. (n) W is the set of all vectors in R4 whose components are all nonzero.
Problem 6.1. In each of the following, a subset W of some n-space R" is de- fined by means of a given condition imposed on the typical vector (₁,...,In). Determine whether W is a subspace of R". (a) W is the set of all vectors in R³ such that 23 = 0. (b) W is the set of all vectors in R³ such that x₁ = 5x2. (c) W is the set of all vectors in R³ such that x₂ = 1. (d) W is the set of all vectors in R³ such that x₁ + x₂ + x3 = 1. (e) W is the set of all vec- in R4 such tors that x1 + 2x2 + 3x3 + 3x4 = 0. (f) W is the set of all vectors in R4 such that 1 3x3 and x₂ = 4x4. (g) W is the set of all vectors in R2 such that r₁| = |x₂|. (h) W is the set of all vectors in R2 such that (1₁)² + (x₂)² = 0. (i) W is the set of all vectors in R2 such that (x₁)² + (x₂)² = 1. (j) W is the set R2 such that of all vectors in ₁| + |₂| = 1. (k) W is the set of all vectors in R such that x₁ + x₂ = x3 + x4. (1) W is the set of all vectors in R4 such that x₁x2 = x3x4- (m) W is the set of all vectors in R4 such that x1x2x3x4 = 0. (n) W is the set of all vectors in R4 whose components are all nonzero.
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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Transcribed Image Text:Problem 6.1. In each of the following, a subset W of some n-space R" is de-
fined by means of a given condition imposed on the typical vector (₁,...,In).
Determine whether W is a subspace of R".
(a) W is the set of all vectors in
R³ such that x3 = 0.
(b) W is the set of all vectors in
R³ such that x₁ = 5x2.
(c) W is the set of all vectors in
R³ such that x₂ = 1.
(d) W is the set of all vectors in
R³ such that x₁ + x₂ + x3 = 1.
(e) W is the set of all vec-
tors in R4 such
x1 + 2x2 + 3x3 + 3x4 0.
that
(f) W is the set of all vectors in
R4 such that ₁ 3x3 and
x₂ = 4x4.
(g) W is the set of all vectors in
R2 such that ₁| = |x₂|.
(h) W is the set of all vectors in
R2 such that (1₁)² + (x₂)² = 0.
(i) W is the set of all vectors in
R2 such that (x₁)² + (x₂)² = 1.
of all vectors in
R2 such that ₁| + |₂| = 1.
(j) W is the set
of all vectors in
R such that x₁ + x₂ = x3 + x4.
(k) W is the set
(1) W is the set of all vectors in
R4 such that ₁2 = x3x4-
(m) W is the set of all vectors in
R4 such that x1x2x3x4 = 0.
(n) W is the set of all vectors in
R4 whose components are all
nonzero.
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