1) W is the set of all vectors from R 4 with components a, b, c, d such that a - 2b + 5c = d and c - a = b. a) Show that W is a subspace of R 4 ( think of the solution space of a homogeneous system in parametric vector form...can you do something like this with these vectors in W?) b) Find a basis for subspace W. c) State the dimension of subspace W and explain how you determined the dimension. 2) Let u =(1,0,0), v = (0,1,0), and W is the set of all vectors (s,s,0) where s is any real number. a) Does span{u,v} = W? Explain why or why not. b) Is {u,v} a basis for W? Explain why or why not.

1) W is the set of all vectors from R 4 with components a, b, c, d such that a - 2b + 5c = d and c - a = b. a) Show that W is a subspace of R 4 ( think of the solution space of a homogeneous system in parametric vector form...can you do something like this with these vectors in W?) b) Find a basis for subspace W. c) State the dimension of subspace W and explain how you determined the dimension. 2) Let u =(1,0,0), v = (0,1,0), and W is the set of all vectors (s,s,0) where s is any real number. a) Does span{u,v} = W? Explain why or why not. b) Is {u,v} a basis for W? Explain why or why not.

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Theorem: Let v1, v2, ...,Vk be vectors in a vector space V. The set of all linear
combinations (which is the span) of those vectors forms a subspace of V.
1) W is the set of all vectors from R 4 with components a, b, c, d such that a - 2b +
5c = d and c - a = b.
a) Show that W is a subspace of R 4
( think of the solution space of a homogeneous system in parametric vector
form...can you do something like this with these vectors in W?)
b) Find a basis for subspace W.
c) State the dimension of subspace W and explain how you determined the
dimension.
2) Let u =(1,0,0), v = (0,1,0), and W is the set of all vectors (s.S,0) where s is any
real number.
a) Does span{u,v}
W? Explain why or why not.
%3|
b) Is {u,v} a basis for W? Explain why or why not.

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