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Mathematics · Advanced university

Dimension of a quotient vector space

W must be a linear subspace of a finite-dimensional vector space V.

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Method and conditions

W must be a linear subspace of a finite-dimensional vector space V.

dim(V/W) = dim V − dim W

Worked examples

Let V be a 30-dimensional vector space and W a 17-dimensional subspace. Find dim(V/W). Round the final result to 4 decimal places if needed.

Extend a basis of W to a basis of V. The 13 added directions form a basis of the quotient.

13

Let V be a 14-dimensional vector space and W a 2-dimensional subspace. Find dim(V/W). Round the final result to 4 decimal places if needed.

Extend a basis of W to a basis of V. The 12 added directions form a basis of the quotient.

12

Try it

  1. Let V be a 33-dimensional vector space and W a 23-dimensional subspace. Find dim(V/W). Round the final result to 4 decimal places if needed.

  2. Let V be a 34-dimensional vector space and W a 16-dimensional subspace. Find dim(V/W). Round the final result to 4 decimal places if needed.

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Answers and marking notes
  1. 10

    Extend a basis of W to a basis of V. The 10 added directions form a basis of the quotient.

  2. 18

    Extend a basis of W to a basis of V. The 18 added directions form a basis of the quotient.

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