Row, Col, Null space

Question 1

What is the basis of a row space of a matrix?

Answer 1

Reduce to row echelon form, the nonzero rows form a bsis for row(A)

Question 2

What is the basis for the colum space of a matrix?

Answer 2

1. Reduce to row echelon form
2. Identify columns containing leading 1s
3. Corresponding columns of original matrix form basis for col(A)

Question 3

What is the null space of a matrix?

Answer 3

All x satisfying Ax = 0

Question 4

How do we find vectors that form a basis for the null space?

Answer 4

1. reduced row echelon form (leading 1s in 0 column)
2. Turn into augmented matrix with 0 appended
3. Solve leading variables with equations
4. Write in parametrised form from general form with free variables
5. The column vectors achieved in parametrised form is the basis

Question 5

What is the row space?

Answer 5

See rows of reduced matrix as separate vectors and take their span

Question 6

What is the column space?

Answer 6

span of column vectors. If linearly independent, the column space spans entire target space.
Otherwise it spans a subspace.

Question 7

What does the following dimensions represent:
- Dim of null
- Dim of col
- Dim row

Answer 7

• null: nullity of matrix / dimension of kernel
• Col: Rank of matrix
• Row: rank of matrix

Question 8

Given that we know the dimension of the column space of a matrix, how can we use the rank nullity theorem to calculate the dimension of the null space?

Answer 8

Dimension of col space = rank
Nullity = dimension of null space
Dim(V) = dimension of matrix (count of columns)
Easy algebra

Question 9

If the det(A) = 0, what does it mean for rank?

Answer 9

A 0 determinant means that the matrix has a full rank.