Question

59. Let U and W be finite dimensional subspaces of a vector space V. If dim U = 2, dim W = 2, dim (U+W) = 3,
then dim (UNW) is
(A) 1
(B) 2
(C) 3
(D) 4

Answer 1

Answer:

59. Let U and W be finite dimensional subspaces of a vector space V. If dim U = 2, dim W = 2, dim (U+W) = 3,

then dim (UNW) is

(A) 1

(B) 2

(C) 3

(D) 4

Step-by-step explanation:

4

Answer 2

The value of dim (U ∩ W) = 1

Given :

• Let U and W be finite dimensional subspaces of a vector space V.

• dim U = 2, dim W = 2, dim (U + W) = 3

To find :

The value of dim (U ∩ W) is

(A) 1

(B) 2

(C) 3

(D) 4

Theorem :

Suppose U and W are finite - dimensional subspaces of a vector space V. Then U + W has finite dimension and dim ( U + W ) = dim U + dim W - dim (U ∩ W)

Solution :

Step 1 of 2 :

Write down the given dimensions

Here it is given that U and W be finite dimensional subspaces of a vector space V.

dim U = 2, dim W = 2, dim (U + W) = 3

Step 2 of 2 :

Find value dim (U ∩ W)

A theorem on vector space that , Suppose U and W are finite - dimensional subspaces of a vector space V. Then U + W has finite dimension and dim ( U + W ) = dim U + dim W - dim (U ∩ W)

Thus we get

dim (U + W) = dim U + dim W - dim (U ∩ W)

⇒ 3 = 2 + 2 - dim (U ∩ W)

⇒ 3 = 4 - dim (U ∩ W)

⇒ dim (U ∩ W) = 4 - 3

⇒ dim (U ∩ W) = 1

Hence the correct option is (A) 1

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