Why is Understanding Column Space and Null Space Important in Linear Algebra?

In summary, understanding column space and null space in linear algebra is important because they provide information about a linear map between vector spaces. The kernel of the map, which is the set of all inputs that map to zero, can be thought of as the null space of the corresponding matrix representation. Similarly, the range of the map, which is the set of all outputs that can be obtained, can be thought of as the column space of the matrix representation. These subspaces can provide valuable insights about the map and its properties.
  • #1
Muthumanimaran
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Why it is important to know about Column space and Null spaces in Linear Algebra?
 
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  • #2
If ##T: X\to Y## is a linear map between vector spaces, then there are a bunch of different reasons to care about the kernel ##\text{ker}T = \{x\in X:\enspace Tx=0\} \subseteq X## and range ##\text{ran}T = \{Tx: \enspace x \in X\} \subseteq Y##. Why/whether we care about those depends on why we care about the map ##T##.

In the special case where ##X## and ##Y## are Euclidean and ##T## is represented by a matrix ##A##, the kernel of ##T## is exactly the null space of ##A##, while the range of ##T## is exactly the column space of ##A##
 
  • #3
Thank you. But I have not done linear mappings yet. I am reading Linear Algebra and its applications by Gilbert strang, 4th edition. while I am reading subspaces (chapter 2) I was wondering what is the use of such subspaces. If you can explain me intuitively without linear mapping it would be very helpful.
 

Related to Why is Understanding Column Space and Null Space Important in Linear Algebra?

What is the column space?

The column space, also known as the range, of a matrix is the span of its column vectors. In other words, it is the set of all possible linear combinations of the columns of the matrix. It represents the space that can be reached by multiplying the matrix with any vector.

What is the null space?

The null space, also known as the kernel, of a matrix is the set of all vectors that when multiplied by the matrix result in a zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the matrix and x is a vector. It represents the space of all vectors that are mapped to the zero vector by the matrix.

How are the column space and null space related?

The column space and null space are orthogonal complements of each other. This means that every vector in the null space is perpendicular to every vector in the column space. In other words, the null space and column space have no common vectors and together span the entire space.

What is the dimension of the column space and null space?

The dimension of the column space is equal to the rank of the matrix, which is the number of linearly independent columns. The dimension of the null space is equal to the number of free variables in the reduced row-echelon form of the matrix. In other words, it is the number of variables that can take any value without affecting the solution to the equation Ax = 0.

How can the column space and null space be used in applications?

The column space and null space are important concepts in linear algebra and have many applications in various fields such as data analysis, coding theory, and computer graphics. They can be used to find a basis for the solution space of a system of linear equations, determine linear independence and dependence of vectors, and perform dimensionality reduction in data analysis.

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