Null space vs Col space dimension?

In summary, the conversation discusses finding integers p and q that make Nul A a subspace of Rp and Col A a subspace of Rq. The participants also question why the values of p and q are different between the two spaces and how the dimension can be 4 when there are only 3 entries in the column vectors. The concept of null space and its relation to the matrix's domain is also mentioned.
  • #1
Shaybay92
124
0
I have a question in my linear algebra text that asks:

Give integers p and q such that Nul A is a subspace of Rp and Col A is a subspace of Rq.


What determines these values? Why are the values of p and q different between the Nul space and Col space? The matrix in question is a 3 x 4 matrix and the value for Col A was 3 and Nul A was 4.

3 2 1 -5
-9 -4 1 7
9 2 -5 1


Why are they different? I would have thought the dimension was just the number of entires in each column. How can Nul space be 4 dimensions when there are only 3 entries in the column vectors?
 
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  • #2
The null space is the kernel of the matrix. What is the domain of your matrix?
 
  • #3
What do you mean by kernel?
 

FAQ: Null space vs Col space dimension?

1. What is the difference between the null space and column space dimensions?

The null space dimension refers to the number of linearly independent vectors that make up the basis for the null space of a matrix. The column space dimension, on the other hand, refers to the number of linearly independent columns in the matrix. In other words, the null space dimension represents the number of vectors that are "lost" in the transformation of the matrix, while the column space dimension represents the number of vectors that are "kept."

2. How do you calculate the dimension of the null space and column space?

The dimension of the null space can be calculated by finding the rank of the matrix, which is the number of linearly independent columns. Then, subtract the rank from the total number of columns in the matrix. The dimension of the column space can be calculated by finding the rank of the matrix, which is the number of linearly independent rows. It is also equal to the number of non-zero rows in the reduced row echelon form of the matrix.

3. Can the null space and column space dimensions be equal?

Yes, it is possible for the null space and column space dimensions to be equal. This happens when there are no "lost" or "kept" vectors in the transformation of the matrix. In other words, when the null space and column space are both zero-dimensional, meaning they only contain the zero vector.

4. How do the null space and column space dimensions relate to the rank of a matrix?

The null space dimension is equal to the number of vectors that are "lost" in the transformation, which is also known as the nullity of the matrix. The column space dimension is equal to the number of vectors that are "kept" in the transformation, which is also known as the rank of the matrix. Therefore, the rank of a matrix plus the nullity of the matrix is equal to the total number of columns in the matrix.

5. Why are the null space and column space dimensions important in linear algebra?

The null space and column space dimensions provide valuable information about the properties and behavior of a matrix. They are used to determine whether a system of linear equations has a unique solution, and to find the basis for the null space and column space. Additionally, the dimensions of the null space and column space are important in determining the invertibility of a matrix and in solving problems related to linear transformations and linear independence.

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