Let V,W be subspaces of the vectorspace U and V Intersection W={0} Show that if v1,v2 is in V and w1,w2 is in W are vectors such that v1+w1=v2+w2 then v1=v2 and w1=w2. Please show all work and explanation. Solution Start with v1+w1=v2+w2 Subtract v2 from both sides to get this: v1+w1-v2=w2 Now subtract w1 from both sides to get this: v1-v2=w2-w1. Since W is a vector space, you know that w2-w1 is in W. Since V is a vector space, you know that v1-v2 is in V. But v1-v2 and w2-w1 equal each other, so that means v1-v2 is in W as well. Since v1-v2 is in both V and W, and since the intersection of V and W is {0}, this means v1- v2=0. Therefore, v1 = v2. Using a similar argument, w2-w1 is in both V and W, and hence w2-w1=0. Therefore, w1 = w2..

1. Let V,W be subspaces of the vectorspace U and V Intersection W={0}
Show that if v1,v2 is in V and w1,w2 is in W are vectors such that v1+w1=v2+w2 then v1=v2
and w1=w2.
Please show all work and explanation.
Solution
Start with v1+w1=v2+w2
Subtract v2 from both sides to get this:
v1+w1-v2=w2
Now subtract w1 from both sides to get this:
v1-v2=w2-w1.
Since W is a vector space, you know that w2-w1 is in W.
Since V is a vector space, you know that v1-v2 is in V.
But v1-v2 and w2-w1 equal each other, so that means v1-v2 is in W as well.
Since v1-v2 is in both V and W, and since the intersection of V and W is {0}, this means v1-
v2=0.
Therefore, v1 = v2.
Using a similar argument, w2-w1 is in both V and W, and hence w2-w1=0.
Therefore, w1 = w2.