Math Is Fun Forum

You are given that A is an 5 x 9 matrix of nullity 5.
1.  The null space of A is a d-dimensional subspace of Rn, where
     d =______and n =_______
2.  The rank of A is______

3.  The row space of A is a d-dimensional subspace of Rn, where
     d =_______and n =________

4.  The column space of A is a d-dimensional subspace of Rn, where
     d =______and n =________   
5.  Are the row vectors of A linearly independent?
6.  Are the column vectors of A linearly independent?

7.  The number of parameters in the solution of ATx = 0 is__________
8.  For every b belongs to R9, the system ATx = b is consistent.

Let A be the matrix
-2 -3 3 -3 -2
-4 -5 2 -8 -6
6 12 -21 1 1
1. A basis for the row space of A is_________
2. Are the row vectors of A linearly independent?
3. A basis for the column space of A is________
4. Are the column vectors of A linearly independent?
5. The rank of A is  and the nullity of A is__________
    The solution set of the system Ax = 0 has  parameters.
6. The rank of AT is  and the nullity of AT is_________
    The solution set of the system ATx = 0 has_______parameters.

You are given that A is an 5 x 9 matrix of nullity 5.
1. The null space of A is a d-dimensional subspace of Rn, where
d =________and n =__________________
2.The rank of A is___________
3.The row space of A is a d-dimensional subspace of Rn, where
d =_______and n =___________
4.The column space of A is a d-dimensional subspace of Rn, where
d =_________and n =_______________
5.Are the row vectors of A linearly independent?
6.Are the column vectors of A linearly independent?
7.The number of parameters in the solution of ATx = 0 is_________________
8.For every b Î R9, the system ATx = b is consistent.

Let A be the matrix
-2 -3 3 -3 -2
-4 -5 2 -8 -6
6 12 -21 1 1

You are asked to find bases for the row and column spaces of A, and to answer a few other questions related to the rank and nullity of A.
1.A basis for the row space of A is_________________
2.Are the row vectors of A linearly independent?____________________
3.A basis for the column space of A is__________________
4.Are the column vectors of A linearly independent?_____________________
5.The rank of A is_________and the nullity of A is_________The solution set of the system Ax = 0 has________parameters.
6.The rank of AT is_________and the nullity of AT is__________The solution set of the system ATx = 0 has_______________parameters.

In R4 let

u1 = (0, 0, 0, -3), u2 = (0, 2, 3, -3), u3 = (-2, -4, -3, -3).
You are asked to determine the dimension of the subspace of R4 spanned by u1, u2, u3, and whether the set {u1, u2, u3} can be extended to a basis for R4.

1.  The set {u1, u2, u3} is linearly independent. 
2.  The dimension of the subspace span{u1, u2, u3} is     
3.  The number of vectors in any basis for R4 is   
4.  Select one of the radio buttons below and provide any additional information requested.
 

The set {u1, u2, u3} cannot be extended to a basis for R4.

The set {u1, u2, u3, w} is a basis for R4, where
w =

Consider the following sets of vectors in R3:

S1 =  { (3, -2, -3), (0, 4, 2), (0, 1, 3) } 
S2 =  { (-2, 3, 4), (2, 4, -3), (4, 3, -3), (4, 3, -2) } 
S3 =  { (0, -3, -4), (0, -12, 0), (0, 6, -12) } 
S4 =  { (-2, 2, 3), (4, -2, 3) } 

You are asked to determine which of these sets are bases for R3. For each of the following statements, select True or False to indicate your answer.   Help

1.  The set S1 is a basis for R3.   
2.  The set S2 is a basis for R3.   
3.  The set S3 is a basis for R3.   
4.  The set S4 is a basis for R3