Null space vs Col space dimension?

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SUMMARY

The discussion centers on the dimensions of the null space (Nul A) and column space (Col A) of a 3 x 4 matrix. The values determined are p = 4 for Nul A and q = 3 for Col A, highlighting that the null space can have a higher dimension than the number of rows in the matrix. This discrepancy arises because the null space represents the solutions to the homogeneous equation Ax = 0, while the column space is defined by the linear combinations of the matrix's columns. Understanding the kernel and the domain of the matrix is crucial for grasping these concepts.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically null space and column space.
  • Familiarity with matrix dimensions and their implications in vector spaces.
  • Knowledge of the kernel of a matrix and its role in linear transformations.
  • Basic understanding of homogeneous equations and their solutions.
NEXT STEPS
  • Study the properties of null space and column space in linear algebra.
  • Learn about the Rank-Nullity Theorem and its applications.
  • Explore the concept of linear transformations and their kernels.
  • Investigate examples of different matrix dimensions and their corresponding null and column spaces.
USEFUL FOR

Students of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of vector spaces and linear transformations.

Shaybay92
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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?
 
Physics news on Phys.org
The null space is the kernel of the matrix. What is the domain of your matrix?
 
What do you mean by kernel?
 

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