# Vector spaces and subspaces

1. Show that the following are vector spaces:
1. The -tuple space, : Let be a Field and let be the set of all -tuples of scalars . If with then their sum is defined as

and the product of a scalar and a vector is

2. The space of matrices, : under usual matrix addition and multiplication of a matrix with a scalar.
3. The space of functions from a set to a Field: under the operations:

and

4. The space of polynomial functions over a Field: with addition and scalar multiplication as defined above.
2. Which of the following sets of vectors in are subspaces of ()?
1. all such that ;
2. all such that ;
3. all such that ;
4. all such that ;
5. all such that is rational.
3. Let be the (real) vector space of all functions . Which of the following are subspaces of ?
1. all such that ;
2. all such that ;
3. all such that ;
4. all such that ;
5. all which are continuous.
4. Let be the set of all which satisfy

Find a finite set of vectors which spans .
5. Let be a Field and let be a positive integer . Let be a vector space of all matrices over . Which of the following set of matrices in are subspaces of ?
1. all invertible ;
2. all non-invertible ;
3. all such that , where is some fixed matrix in ;
4. all such that .
1. Prove that the only subspaces of are and the zero subspace.
2. Prove that a subspace of is , or the zero subspace, or consists of all scalar multiples of some fixed vector in .
3. What can you say about the subspaces of ?
6. Let be a matrix over . Show that the set of all vectors such that is a subspace of . This subspace is called the null space of and its dimension in the nullity of .
7. Let be a matrix over . Show that the set of all vectors spanned by the row vectors of is a subspace of . This subspace is called the row space of and its dimension in the row rank of .
8. Let be a matrix over . Show that the set of all vectors such that has a solution for is a subspace of . This subspace is called the range space of and its dimension in the column rank of (why?).
9. Show that for any matrix
1. nullity + row rank =
2. nullity + column rank =
and conclude that
row rank of = column rank of
10. Consider the matrix

1. Find an invertible matrix such that is a row-reduced echelon matrix .
2. Find a basis for the row space of .
3. Say which vectors are in .
4. Find the coordinate matrix of each vector in the ordered basis chosen in (b).
5. Write each vector as a linear combination of the rows of .
6. Give an explicit description of the null space of .
7. Find a basis for the null space.
8. For what column matrices does the equation have solutions ?
9. Explicitly find the range space of and find a basis.
10. Verify the relations regarding the nullity, row rank and column rank of .
Subhashis Banerjee 2009-09-03