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Mathematics · University

Bases, rank and linear maps

A basis both spans a vector space and is linearly independent.

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Prerequisites

Systems of linear equations and vector arithmetic.

Study notes

Basis

A basis both spans a vector space and is linearly independent.

Rank

The rank is the dimension of the image, or equivalently the number of pivot columns.

Nullity

Rank plus nullity equals the dimension of the domain.

Subspace test

A candidate subspace must contain zero and be closed under addition and scalar multiplication.

Worked examples

A linear map from R^7 to R^5 has rank 4. Find its nullity.

Rank-nullity uses the 7-dimensional domain: 7−4=3.

3

Show that vectors (1,0) and (1,1) form a basis of R².

Credit spanning and independence, or a nonzero determinant argument.

A linear combination a(1,0)+b(1,1)=(a+b,b) can represent every (x,y) using b=y,a=x−y. The zero combination forces a=b=0.

Check your understanding

  1. Which set is a linear subspace of R²?

    1. The unit circle
    2. All (x,y) with x+y=0
    3. All points with x>0
    4. All (x,y) with x+y=1

  2. What is the rank of [[1,2],[2,4]]?

  3. Can three vectors be linearly independent in R²?

    1. Yes, if all entries are nonzero
    2. Yes, if all lengths differ
    3. Only when perpendicular in pairs
    4. No

  4. Explain why the pivot columns of a row-reduced matrix identify basis columns in the original matrix rather than replacing them.

Answers and marking guidance

Answers and marking notes
  1. All (x,y) with x+y=0

    The homogeneous solution set contains zero and is closed under linear combinations.

  2. 1

    The second row is twice the first and the matrix is not zero.

  3. No

    No independent set has more vectors than the dimension.

  4. Row operations preserve linear relations among columns, so pivot positions select independent original columns spanning the original column space. The reduced columns may live in a different column space.

    Credit the distinction between selecting original columns and using transformed columns.

Further reading and review references

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