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defined as Let V = {(x,y) | x, y = R}, with vector addition and scalar multiplication (x₁, y₁) (x2, y2) = (₁ + x2, 3₁+ y2 + 1) and a Ⓒ (x, y) = (ar, ay + a-1). (Recall from lecture that the zero vector is (0, -1) in this vector space.) (a) Show that {(1,0), (0, 1)} is a linearly independent set of vectors in V. (b) Show that {(1, 0), (0, 1)} is a basis of V. (c) Given any vector (x, y) EV, write (x, y) as a linear combination of (1,0) and (0, 1).

defined as Let V = {(x,y) | x, y = R}, with vector addition and scalar multiplication (x₁, y₁) (x2, y2) = (₁ + x2, 3₁+ y2 + 1) and a Ⓒ (x, y) = (ar, ay + a-1). (Recall from lecture that the zero vector is (0, -1) in this vector space.) (a) Show that {(1,0), (0, 1)} is a linearly independent set of vectors in V. (b) Show that {(1, 0), (0, 1)} is a basis of V. (c) Given any vector (x, y) EV, write (x, y) as a linear combination of (1,0) and (0, 1).

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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. Let V = {(x, y) | x, y ER}, with vector addition and scalar multiplication defined as (x₁, y₁) (T2, y2) = (x₁+x2, Y1 + y2 + 1) and a Ⓒ (x, y) = (ax, ay+a-1). (Recall from lecture that the zero vector is (0, -1) in this vector space.) (a) Show that {(1, 0), (0, 1)} is a linearly independent set of vectors in V. (b) Show that {(1,0), (0, 1)} is a basis of V. (c) Given any vector (x, y) € V, write (x, y) as a linear combination of (1,0) and (0, 1).
Transcribed Image Text:. Let V = {(x, y) | x, y ER}, with vector addition and scalar multiplication defined as (x₁, y₁) (T2, y2) = (x₁+x2, Y1 + y2 + 1) and a Ⓒ (x, y) = (ax, ay+a-1). (Recall from lecture that the zero vector is (0, -1) in this vector space.) (a) Show that {(1, 0), (0, 1)} is a linearly independent set of vectors in V. (b) Show that {(1,0), (0, 1)} is a basis of V. (c) Given any vector (x, y) € V, write (x, y) as a linear combination of (1,0) and (0, 1).
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