2 Direct Sum
Let U and W be subspaces of the vector space V. In this project you will explore the sum and direct sum of subspaces.
Define the sum of the subspaces U and W as follows :
I need help with only the first three questions
2
Direct Sum
sum of subspaces
1.Define the sum of the subspaces U and W as follows :
U+W={u+w:uEUandwEW}
Prove that U+W is subspace of V.
2. Consider the subspaces of R3 listed below
U={(x,y,x-y):x,yeR} W={x,0,x)xeR} Z={(x,x,x):xeR}
FindU+WU+Z,andW+Z
3. If U and W are subspaces of V such that V=U+W and U NW={0},prove that every vector in V has a unique representation of the form u +w,where u U and w E W. In this case,we say that V is a direct sum of U and Wand write v=uW Which of the sums in question (2) of this project are direct sums? 4. Let V =U@W and suppose that {u1,U2,...,uk} is a basis for the subspaceU, and {w,W2,...,Wm} is a basis for the subspace W. Prove that {u1, U2,..., Uk, W1, W2,... , Wm} is a basis for the sub- space V. 5.Consider the subspaces of V=R3 listed below U={x,0,y:x,yeR} W={(0,x,y:x,yER}
a.Show that R3=U+W. b.Is R3 a direct sum of U and W? c.What are the dimensions of UW,UW,and U+W? d.In general,formulate a conjecture that relates the dimensions of U,W,UW,and U+W 6. Can you find two 2-dimensional subspaces of R3 whose intersection is just the zero vector? Why or why not ?
