Linear algebra 1. The given set with the given operations is not a vector space. Check all the properties and list whichproperties fail to hold.The set of all ordered pairs of real numbers with the operations (x,, y,) O (x,, y2) = (x, +x,,y, + y,) andrO(x, y) = (0,0).2. Prove that a vector u in a vector space has only one negative -u. [Hint: Assume that there are twonegatives and conclude that they must be equal.]3. C(-00, 00) denotes the vector space of real-valued continuous functions.Which of the following subsets are subspaces of the vector space C(-x,0) ? Explain.a) All nonnegative functions.b) All constant functions.c) All functions f such that f(0) = 0.d) All functions f such that f(0) = 5.e) All differentiable functions.4. a) Prove that proj (proj-v) = proj,vb) Prove that proj (v – proj v) =0.c) Explain parts a) and b) geometrically.5. a) Prove that u-v2u + v for all vectors in R".b) Explain parts a) geometrically.

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Answered: 1. The given set with the given…

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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Linear algebra

1. The given set with the given operations is not a vector space. Check all the properties and list which
properties fail to hold.
The set of all ordered pairs of real numbers with the operations (x,, y,) O (x,, y2) = (x, +x,,y, + y,) and
rO(x, y) = (0,0).
2. Prove that a vector u in a vector space has only one negative -u. [Hint: Assume that there are two
negatives and conclude that they must be equal.]
3. C(-00, 00) denotes the vector space of real-valued continuous functions.
Which of the following subsets are subspaces of the vector space C(-x,0) ? Explain.
a) All nonnegative functions.
b) All constant functions.
c) All functions f such that f(0) = 0.
d) All functions f such that f(0) = 5.
e) All differentiable functions.
4. a) Prove that proj (proj-v) = proj,v
b) Prove that proj (v – proj v) =0.
c) Explain parts a) and b) geometrically.
5. a) Prove that u-v2u + v for all vectors in R".
b) Explain parts a) geometrically.

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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