Addition of Subspaces: meaning of '(A+B) intersect C'
- Thread starter moonbeam
- Start date
For subspaces \(\displaystyle A\) and \(\displaystyle B\) of \(\displaystyle \mathbb{R}^3\), \(\displaystyle A + B\) is the set of all vectors in \(\displaystyle \mathbb{R}^3\) of the form \(\displaystyle \vec{v} + \vec{w}\) where \(\displaystyle \vec{v}\) is in \(\displaystyle V\) and \(\displaystyle \vec{w}\) is in \(\displaystyle W\).
It means "The set of vectors in C which can be written as \(\displaystyle a+b\) for some \(\displaystyle a \in A, \,\, b\in B\)."
Example (k,l,m arbitrary integers):
\(\displaystyle A=\{a \in R^3; \,\, a=(3k,5l,7m)\}\)
\(\displaystyle B=\{b \in R^3; \,\, b=(2k,4l,6m)\}\)
\(\displaystyle C=\{c \in R^3; \,\, c=(7k,11l,13m)\}\)
Then \(\displaystyle (A+B)\cap C\) will be the set of all \(\displaystyle (x,y,z)\) such that x is a sum of a multiple of 3 and a multiple of 2 and is also a multiple of 7. Similarly, y would have to be the sum of a multiple of 5 and a multiple of 4 and also be a multiple of 11. And similarly for z.
i.e.
\(\displaystyle (A+B)\cap C \,\, = \,\, \{(x,y,z); \ \,\, x=(3a+2b)\,\, and \,\, x=7c \\ \,\,\,\,\,\, \,\, \,\, \,\,\,\, \,\, y=(5d+4e)\,\, and \,\, y=11l \\ \,\,\,\, \,\, \,\,\,\, \,\, \,\, \,\, z=(7g+6h)\,\, and \,\, z=13i \\ \,\,\,\, \,\, \,\, \,\, \,\, \,\, \,\, \text{for some a,b,c,d,e,f,g,h,i \in Z}\}\)
I hope that made sense.
Example (k,l,m arbitrary integers):
\(\displaystyle A=\{a \in R^3; \,\, a=(3k,5l,7m)\}\)
\(\displaystyle B=\{b \in R^3; \,\, b=(2k,4l,6m)\}\)
\(\displaystyle C=\{c \in R^3; \,\, c=(7k,11l,13m)\}\)
Then \(\displaystyle (A+B)\cap C\) will be the set of all \(\displaystyle (x,y,z)\) such that x is a sum of a multiple of 3 and a multiple of 2 and is also a multiple of 7. Similarly, y would have to be the sum of a multiple of 5 and a multiple of 4 and also be a multiple of 11. And similarly for z.
i.e.
\(\displaystyle (A+B)\cap C \,\, = \,\, \{(x,y,z); \ \,\, x=(3a+2b)\,\, and \,\, x=7c \\ \,\,\,\,\,\, \,\, \,\, \,\,\,\, \,\, y=(5d+4e)\,\, and \,\, y=11l \\ \,\,\,\, \,\, \,\,\,\, \,\, \,\, \,\, z=(7g+6h)\,\, and \,\, z=13i \\ \,\,\,\, \,\, \,\, \,\, \,\, \,\, \,\, \text{for some a,b,c,d,e,f,g,h,i \in Z}\}\)
I hope that made sense.