Question

Rove that in any vector space V(α+β)(u+ v )=αx+αy +βx +βy

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for any α,β∈R and any
u , v∈V

Answer 1

To prove the equation (α+β)(u+ v )=αx+αy +βx +βy, expand and rearrange using the distributive property of scalar multiplication over vector addition.

Step-by-step explanation:

To prove that in any vector space V, (α+β)(u+ v )=αx+αy +βx +βy, for any α,β∈R and any u, v∈V, we can use the distributive property of scalar multiplication over vector addition. Let's start by expanding the left side of the equation:

(α+β)(u+ v) = α(u + v) + β(u + v)

Next, we can use the distributive property to expand each term:

α(u + v) + β(u + v) = αu + αv + βu + βv

We can now rearrange the terms on the right side of the equation:

αu + αv + βu + βv = αx + αy + βx + βy

Since the left side and right side of the equation are equal, we have proven that (α+β)(u+ v )=αx+αy +βx +βy for any α,β∈R and any u, v∈V.