Column And Null Space Calculator

Matrix Space Calculations:

\[ \text{Col}(A) = \text{Span of column vectors of } A \] \[ \text{Nul}(A) = \{ x \in \mathbb{R}^n \mid Ax = 0 \} \]

Column Space Basis:

Null Space Basis:

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1. What is Column Space?

The column space of a matrix A, denoted as Col(A), is the set of all possible linear combinations of its column vectors. It represents the range of the linear transformation defined by the matrix.

2. What is Null Space?

The null space of a matrix A, denoted as Nul(A), is the set of all vectors x such that Ax = 0. It represents the solutions to the homogeneous system of linear equations.

3. How the Calculator Works

The calculator performs matrix operations to determine:

\[ \text{Col}(A) = \text{Span}\{a_1, a_2, ..., a_n\} \] \[ \text{Nul}(A) = \{ x \mid Ax = 0 \} \]

Where:

\( A \) — Input matrix
\( a_1, a_2, ..., a_n \) — Column vectors of A
\( x \) — Solution vector to homogeneous system

4. Using the Calculator

Instructions: Enter your matrix with rows separated by newlines and elements separated by spaces. The calculator will compute both column space and null space bases.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between column space and null space?
A: Column space represents where the matrix can map vectors to, while null space represents what vectors get mapped to zero.

Q2: How are these spaces related to rank and nullity?
A: Rank = dimension of column space, Nullity = dimension of null space. Rank + Nullity = number of columns.

Q3: What does a zero null space indicate?
A: A zero null space means the matrix is injective (one-to-one) and has full column rank.

Q4: Can these spaces be calculated for non-square matrices?
A: Yes, both column space and null space can be computed for any m×n matrix.

Q5: What practical applications do these concepts have?
A: Used in solving linear systems, understanding linear transformations, data analysis, and many engineering applications.