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Definition Example Given a set {x₁,..., xk} of vectors in R", we let the contained span (or c-span) of those vectors be the set of all contained linear combinations of those vectors: c-span {x₁,..., xk} = {a₁x₁ + +akxk|0 ≤a₁,..., an ER, and a₁ + ... =1} + an - For any single vector v E R", c-span {v} is just the set {v}. This is because if we consider av and apply the conditions for a contained linear combination, we see that a must equal 1! Note, therefore, that c-span {v} will only be a subspace if v = 0. 3.1 Suppose that u and v are non-parallel, non-zero vectors in R². You will show that c-span {u, v} is the line segment between U and V (the tips of u and V, respectively). To do this, complete the following: (a) Let d = UV, and consider the line L given by x = u + td, which U and V lie on. Show that if we c-span {u, v}, then the point W (which w points at) is on the line L. Hint: rewrite one of u or v in terms of d and the other vector, then rearrange the result to show that w must therefore satisfy the equation of L. (b) Now, suppose that w points to a point W on L which is not on the line segment from U to V. Show that w is not in c-span{u, v}. Hint: You may assume that if a vector (w here) is written in one way as an (ordinary) linear combination of u and v, then there is no other way to write it as a linear combination of u and v. 1 (c) Finally, show that if W is on the line segment from U to V, then w must be in c-span{u, v}. You are allowed to argue for parts (b) and (c), or even all of (a), (b) and (c) together if you want.

Definition Example Given a set {x₁,..., xk} of vectors in R", we let the contained span (or c-span) of those vectors be the set of all contained linear combinations of those vectors: c-span {x₁,..., xk} = {a₁x₁ + +akxk|0 ≤a₁,..., an ER, and a₁ + ... =1} + an - For any single vector v E R", c-span {v} is just the set {v}. This is because if we consider av and apply the conditions for a contained linear combination, we see that a must equal 1! Note, therefore, that c-span {v} will only be a subspace if v = 0. 3.1 Suppose that u and v are non-parallel, non-zero vectors in R². You will show that c-span {u, v} is the line segment between U and V (the tips of u and V, respectively). To do this, complete the following: (a) Let d = UV, and consider the line L given by x = u + td, which U and V lie on. Show that if we c-span {u, v}, then the point W (which w points at) is on the line L. Hint: rewrite one of u or v in terms of d and the other vector, then rearrange the result to show that w must therefore satisfy the equation of L. (b) Now, suppose that w points to a point W on L which is not on the line segment from U to V. Show that w is not in c-span{u, v}. Hint: You may assume that if a vector (w here) is written in one way as an (ordinary) linear combination of u and v, then there is no other way to write it as a linear combination of u and v. 1 (c) Finally, show that if W is on the line segment from U to V, then w must be in c-span{u, v}. You are allowed to argue for parts (b) and (c), or even all of (a), (b) and (c) together if you want.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 2CM: Take this test to review the material in Chapters 4and Chapters 5. After you are finished, check...
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Given a set {x1,..., X} of vectors in R", we let the contained span (or c-span) of those vectors be the set of all contained linear combinations of those vectors: C-span{x1,..., xk} = {a1x1 + ... +ajXx | 0 < a1,., an E R, and a1 +... + an = 1} ••. For any single vector v E R", c-span{v} is just the set {v}. This is because if we consider av and apply the conditions for a contained linear combination, we see that a must equal 1! Note, therefore, that c-span{v} will only be a subspace if v = = 0. 3.1 Suppose that u and v are non-parallel, non-zero vectors in R2. You will show that C-span{u, v} is the line segment between U and V (the tips of u and v, respectively). To do this, complete the following: (a) Let d = UV, and consider the line L given by x = u + td, which U and V lie on. Show that if w E c-span{u,v}, then the point W (which w points at) is on the line L. Hint: rewrite one of u or v in terms of d and the other vector, then rearrange the result to show that w must therefore satisfy the equation of L. (b) Now, suppose that w points to a point W on L which is not on the line segment from U to V. Show that w is not in c-span{u,v}. Hint: You may assume that if a vector (w here) is written in one way as an (ordinary) linear combination of u and v, then there is no other way to write it as a linear combination of u and v. 1 (c) Finally, show that if W is on the line segment from U to V, then w must be in C-span{u, v}. You are allowed to argue for parts (b) and (c), or even all of (a), (b) and (c) together if you want. Example Definition
Transcribed Image Text:Given a set {x1,..., X} of vectors in R", we let the contained span (or c-span) of those vectors be the set of all contained linear combinations of those vectors: C-span{x1,..., xk} = {a1x1 + ... +ajXx | 0 < a1,., an E R, and a1 +... + an = 1} ••. For any single vector v E R", c-span{v} is just the set {v}. This is because if we consider av and apply the conditions for a contained linear combination, we see that a must equal 1! Note, therefore, that c-span{v} will only be a subspace if v = = 0. 3.1 Suppose that u and v are non-parallel, non-zero vectors in R2. You will show that C-span{u, v} is the line segment between U and V (the tips of u and v, respectively). To do this, complete the following: (a) Let d = UV, and consider the line L given by x = u + td, which U and V lie on. Show that if w E c-span{u,v}, then the point W (which w points at) is on the line L. Hint: rewrite one of u or v in terms of d and the other vector, then rearrange the result to show that w must therefore satisfy the equation of L. (b) Now, suppose that w points to a point W on L which is not on the line segment from U to V. Show that w is not in c-span{u,v}. Hint: You may assume that if a vector (w here) is written in one way as an (ordinary) linear combination of u and v, then there is no other way to write it as a linear combination of u and v. 1 (c) Finally, show that if W is on the line segment from U to V, then w must be in C-span{u, v}. You are allowed to argue for parts (b) and (c), or even all of (a), (b) and (c) together if you want. Example Definition
We say that a linear combination a1x1+... + aµX½ (with a; e R) of vectors X1,..., X E R" is a contained linear combination if all the a; are non-negative (i.e. a; > 0) and a1 + ... + an = 1. Definition
Transcribed Image Text:We say that a linear combination a1x1+... + aµX½ (with a; e R) of vectors X1,..., X E R" is a contained linear combination if all the a; are non-negative (i.e. a; > 0) and a1 + ... + an = 1. Definition
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