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(b) Let v₁,..., Um be a list of vectors in a linear space. Define what is understood by saying that {v₁,...,Um} is a basis of V and what is understood by the dimension of V.

(b) Let v₁,..., Um be a list of vectors in a linear space. Define what is understood by saying that {v₁,...,Um} is a basis of V and what is understood by the dimension of V.

Advanced Engineering Mathematics
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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PLEASE ANSWER ALL PARTS OF QUESTION CLEARLY

(b) Let v₁,...,Um be a list of vectors in a linear space. Define what is understood by saying that {v₁,...,Um} is a basis of V and what is understood by the dimension of V. azt³ + (c) Let P3 denote the space of all polynomials, p, of the form p(t) a₂t² + a₁t + ao where aj € R for j = 0, 1, 2, 3. The set {t³, t², t, 1} is a basis of P3. Determine which of the following sets of vectors are also bases of P3. (i) {t³,t³ + t², t³ + t² + t, t³ + t² + t + 1} (ii) {t³ + 2t - 4, −2t + 11, t³ + t² +t+1}. = (d) Let V₁, V2, V3 be vectors in a linear space V. Suppose that {v₁, v₂} is linearly independent and that v3 does not belong to the span of v₁ and v₂. Show that {V₁, V2, V3} is linearly independent. Hint: Suppose that a, b and c are such that av₁ + bv₂ + cv3 = 0 and show that a = b = c = 0.
Transcribed Image Text:(b) Let v₁,...,Um be a list of vectors in a linear space. Define what is understood by saying that {v₁,...,Um} is a basis of V and what is understood by the dimension of V. azt³ + (c) Let P3 denote the space of all polynomials, p, of the form p(t) a₂t² + a₁t + ao where aj € R for j = 0, 1, 2, 3. The set {t³, t², t, 1} is a basis of P3. Determine which of the following sets of vectors are also bases of P3. (i) {t³,t³ + t², t³ + t² + t, t³ + t² + t + 1} (ii) {t³ + 2t - 4, −2t + 11, t³ + t² +t+1}. = (d) Let V₁, V2, V3 be vectors in a linear space V. Suppose that {v₁, v₂} is linearly independent and that v3 does not belong to the span of v₁ and v₂. Show that {V₁, V2, V3} is linearly independent. Hint: Suppose that a, b and c are such that av₁ + bv₂ + cv3 = 0 and show that a = b = c = 0.
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