Step 1: General form for vectors in U_2
Since U_2 is not given explicitly in the exercise, we assume that U_2 refers to U. In this case, vectors in U have the property b + c + d = 0, so we can rewrite any vector in U as (a, b, -b-d, d).
Step 2: Finding a basis for U
A basis for U can be found by taking the linearly independent vectors formed from the general form of a vector in U. Let the basis vectors be u_1, u_2, and u_3:
\(u_1 = (1, 0, 0, 0)\),
\(u_2 = (0, 1, -1, 0)\),
\(u_3 = (0, 0, -1, 1)\).
These vectors are linearly independent and span U. Therefore, the basis for U (or U_2) is {u_1, u_2, u_3}.
Step 3: Dimension of U_2
Since U (or U_2) has a basis consisting of three vectors, the dimension of U_2 is 3.
(ii) Finding the basis and dimension of W
Step 4: General form for vectors in W
Vectors in W have the properties a + b = 0 and c = 2d, so we can rewrite any vector in W as (-b, b, 2d, d).
Step 5: Finding a basis for W
A basis for W can be found by taking the linearly independent vectors formed from the general form of a vector in W. Let the basis vectors be w_1 and w_2:
\(w_1 = (-1, 1, 0, 0)\),
\(w_2 = (0, 0, 2, 1)\).
These vectors are linearly independent and span W. Therefore, the basis for W is {w_1, w_2}.
Step 6: Dimension of W
Since W has a basis consisting of two vectors, the dimension of W is 2.
(iii) Finding the basis and dimension of U ∩ W
Step 7: Intersection of U and W
In the intersection of U and W, we will have vectors (a, b, c, d) that satisfy both the properties of U and W, i.e., b + c + d = 0 and a + b = 0, c = 2d.
Step 8: Finding a basis for U ∩ W
From the general form of vectors in U and W, we have
U: (a, b, -b-d, d),
W: (-b, b, 2d, d).
Set the general forms equal to each other and solve for a, b, c, and d:
\((-b, b, 2d, d) = (a, b, -b-d, d)\)
We get
a = -b, c = 2d.
So, the vector in U ∩ W is in the form (-b, b, 2d, d). We can rewrite it as v = b(-1, 1, 0, 0) + d(0, 0, 2, 1) = b w_1 + d w_2, where w_1 and w_2 are already the basis vectors for W.
Since the vector v can be written as a linear combination of the basis vectors of W, the intersection U ∩ W has the same basis as W, which is {w_1, w_2}.
Step 9: Dimension of U ∩ W
Since U ∩ W has a basis consisting of two vectors, the dimension of U ∩ W is 2.
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