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5. Let U, W be subspaces of a vector space V such that (v, ) is a basis for U and (v2, es, 4) is a basis for W, then dim(U + W)S4

5. Let U, W be subspaces of a vector space V such that (v, ) is a basis for U and (v2, es, 4) is a basis for W, then dim(U + W)S4

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Exercise 5: Prove the following claims: 1. If S is a linearily independent set and TCS then T is linearily independent 2. Let v, , t3, e, be vectors such that v E span{v, y, v3} then span{v1, , t3, t4} = span{r, 2, t} 3. Any two scalar 5 x 5 matrices are linearily dependent. 4. For any two vecetors u, v in a vector space V, {u, v} is linearily independent if and only if {u + v, u - v} is linearily independent. 5. Let U, W be subspaces of a vector space V such that (v1, e2) is a basis for U and (v2, ea, 4} is a basis for W, then dim(U + W)<4
Transcribed Image Text:Exercise 5: Prove the following claims: 1. If S is a linearily independent set and TCS then T is linearily independent 2. Let v, , t3, e, be vectors such that v E span{v, y, v3} then span{v1, , t3, t4} = span{r, 2, t} 3. Any two scalar 5 x 5 matrices are linearily dependent. 4. For any two vecetors u, v in a vector space V, {u, v} is linearily independent if and only if {u + v, u - v} is linearily independent. 5. Let U, W be subspaces of a vector space V such that (v1, e2) is a basis for U and (v2, ea, 4} is a basis for W, then dim(U + W)<4
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