Question on Null and Image Space matrices wee

In summary, the conversation discusses finding the Null space and image space of a matrix. The original matrix is row reduced to a square matrix, with a column of 0's added. The Null space is found to be a set of vectors, including one with all 0's and one with a 1. There is confusion about why the vector with all 0's is also included in the Null space. The conversation also mentions finding the image space, but there is uncertainty about how to proceed after setting variables equal to r, s, t, and u. The correctness of the image space found in a separate problem is also questioned.
  • #1
mr_coffee
1,629
1
Hello everyone! I'm alittle confused, my professor found the Null space after some students didn't get how to do it, such as myself. Here is the image and problem:
http://img497.imageshack.us/img497/4161/lastscan9es.jpg
Thats the row reduced forum of the orginal matrix. The orginal matrix was:
-1 4 1
7 -9 0
10 3 -3
-9 1 2

he added a column of 0's so it wuold make it a square matrix, then he row reduced to get the matrix in the link.
I understand where the Null space of A is:
9/37
7/37
1
0

that makes sense to me, but then he adds in:
d =
0
0
0
1
why is that also a set of vectors in the Null space? d is just all 0's, how can he have a 1 also?


Then I'm lost on what I'm suppose to do to find the image space, you can see my work in the image above. What do i do after i set them equal to r,s,t,u? Thanks! :smile:
 
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  • #2
I was going to make a new thread, but its the same question different problem. Did I find the correct image space on this problem?
http://img500.imageshack.us/img500/2195/lastscan7tn.jpg
THanks!
 
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Related to Question on Null and Image Space matrices wee

1. What is the purpose of a Null Space matrix?

The Null Space matrix, also known as the Kernel matrix, is used to find the solutions to a system of linear equations that result in a zero vector. It helps to identify the vectors that will produce a zero output when multiplied by the matrix.

2. How is the Image Space matrix different from the Null Space matrix?

The Image Space matrix, also called the Range matrix, is used to find the output vectors produced by a given set of input vectors when multiplied by the matrix. It represents the span of all possible outputs. The Null Space matrix, on the other hand, represents the subspace of all possible inputs that result in a zero output.

3. Can the Null Space and Image Space matrices have the same dimensions?

No, the Null Space and Image Space matrices cannot have the same dimensions. The Null Space matrix has dimensions of m x n, where m is the number of rows and n is the number of columns, while the Image Space matrix has dimensions of n x p, where p is the number of columns. In order for the matrices to have the same dimensions, n would have to be equal to both m and p, which is not possible.

4. How are Null Space and Image Space matrices used in linear algebra?

Null Space and Image Space matrices are used to understand the properties of a linear transformation and to solve systems of linear equations. They also play a crucial role in determining the rank and nullity of a matrix, which are important concepts in linear algebra.

5. What is the relationship between the Null Space and Image Space matrices?

The Null Space and Image Space matrices are complementary to each other. The dimensions of their intersection are always zero, and together they span the entire vector space. This means that every vector in the vector space can be represented as a sum of vectors in the Null Space and Image Space.

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