We have shown that if there is no number C such that cA = B, Then A and B form a basis of R^2, and that R^2 is a direct sum of the subspaces generated by A and B respectively. We also proved the last assertion of the section concerning the dimensions of U x W. Finally, We found a basis of U x W IF { Fn, Uh } is a basis of U and (W, w ) is a basis of W.
Therefore, the basis for U × W is the set of all ordered pairs (uᵢ, wⱼ) where uᵢ is in U and wⱼ is in W.
- Step-by-step explanation DEFINE THE PROBLEM:
(1) - We need to show that if there is no number C such that cA = B, Then, A and B form a basis of R^2, and that R^2 is a direct sum of the subspaces generated by A and B respectively.
We also need to prove the last assertion of the section concerning the dimension of U x W.
Finally, We need to find a basis of U x W if {Fn, Uh } is a basis of U and (W, w ) is a basis of W.
- (2) - SHOW THAT A and B form a BASIS of R^2:
Since there is no number C such that cA = B, It means that A and B are Linearly Independent. In R^2, If we have Two Linearly Independent Vectors, They Form a Basis. Therefore, A and B form a Basis of R^2.
- (3) - SHOW THT R^2 is a DIRECT SUM of the SUBSPACES GENERATION BY A and B:
Since A and B Form a Basis of R^2, Any Vector in R^2 can be written as a LINEAR COMBINATION of A and B. Also, Since A and B are LINEARLY INDEPENDENT, The Representation of any vector in R^2 as a LINEAR COMBINATION of A and B is UNIQUE. Therefore, R^2 is a DIRECT SUM of the SUBSPACES GENERATED by A and B.
(4) - PROVE THE LAST ASSERTION OF THE SECTION CONCERNING THE DIMENSION OF U x W:
The DIMENSION of U x W is EQUAL to the SUM of the DIMENSIONS of U x W. This is because, Each ELEMENT of U x W can be REPRESENTED as a PAIR of ELEMENTS from U and W, and the representation is UNIQUE.
- (5) - FIND A BASIS OF U x W IF { Fn, Uh } is a basis of U and (W, w ) is a basis of W:
A basis of U x W can be formed by taking all possible pairs of the basis elements from U and W. In this case, The basis of U x W is
U x W is: {( Fn, W ), (Fn, W ), (uh, W ), (uh, W )}.
- (6) - DRAW THE CONCLUSION:
We have shown that if there is no number C such that cA = B, Then A and B form a basis of R^2, and that R^2 is a direct sum of the subspaces generated by A and B respectively. We also proved the last assertion of the section concerning the dimensions of U x W. Finally, We found a basis of U x W IF { Fn, Uh } is a basis of U and (W, w ) is a basis of W. Therefore, the basis for U × W is the set of all ordered pairs (uᵢ, wⱼ) where uᵢ is in U and wⱼ is in W.
I hope this helps you!