Chapter 4.8

Question 1

Theorem of the dimensions of the row and column space

Answer 1

dim(row space) = dim(column space)

Question 2

Def: Rank and Nullity

Answer 2

The common dimension of the column and row space of a matrix a is called the rank of A and is denoted rank(A); the dimension of the null space of A is called the nullity of A and is denoted null(A)

Question 3

Dimension theorem for matrices

Answer 3

If A is a matrix with n columns, then

rank(A) + null(A) = n

Question 4

What relationship is there between rank, nullity and Ax=0

Answer 4

If A is an mxn matrix then

1) rank(A) = The number of leading veriables in the general solution of Ax=0
2) null(A) = The number of free variables in the general solution of Ax=0

Question 5

Number of parameters to the general solution to Ax=b where A is an mxn matrix?

Answer 5

n-rank(A)

Question 6

Theorem of overdetermined and underdetermined case

Answer 6

1) Overdetermined case: If m>n, then the system is inconsistemt for at least one vector b inR^(n)
2) Underdetermined case: If m<n, then for each vector b in R^(m) the linear system Ax=b is either inconsistent or has infinately many solutions.

Question 7

What are the fundamental spaces of a matrix A?

Answer 7

Row space of A, column space of A, null space of A and null space of A^(t)

Question 8

What can be said about the column and row space of A and A^(t)

Answer 8

The row space of A is the column space of A^(t)

The column space of A is the row space of A^(t)

Question 9

If A is an mxn matrix, what can be said about the row column and null space in relation to m, n and r? (3)

Answer 9

1) dim(row(A)) = dim(col(A)) = rank(A) = r
2) dim(null(A^(t)))= m-r
3) r + dim(null(A^(t))) = m

Question 10

Def Orthogonal complement

Answer 10

If W is a subspace of R^(n), then the set of all vectors in R^(n) that is orthogonal to every vector in W is called the orthogonal complement of W and is denoted by the symbol W^(⊥)

Question 11

If W is a subspace of R^(n) then: (W^(⊥)) (3)

Answer 11

1) W^(⊥) is a subspace of R^(n)
2) The only common vector to W and W^(⊥) is {0}
3) The orthogonal complement of W^(⊥) is W

Question 12

If A is an mxn matrix then (null, vector, row space and W^(⊥)) (2)

Answer 12

1) The null space of A and the row space of A are orthogonal complements in R^(n)
2) The null space of A^(T) and the column space of A are orthogonal complements in R^(m)

Question 13

Equivalent statements: If A is an nxn matrix, then the following statements are equivalent.

Answer 13

1) A is invertible
2) Ax=0 only has the trivial solution
3) A is expressable as a product of elementary matrices.
4) The reduced row echelon form of A is I
5) Ax=b is consistent for every nx1 matrix b
6) Ax=b has exactly one solution for every nx1 matrix b
7) det(A)=0
8) The column vectors of A are linearly independent.
9) The row vectors of A are linearly independent
10) The column vectors of A span R^(n)
11) The row vectors of A span R^(n)
12) The column vectors of A form a basis for R^(n)
13) The row vectors of A form a basis for R^(n)
14) A has rank n
15) A has nullity 0
16) The orthogonal complement of the nullspace of A is R^(n)
17) The orthogonal complement of the row space of A is {0}