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Enter Your Matrix
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3D Visualization
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Column C(A)
Null N(A)
Row C(AT)
Left Null N(AT)
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Matrix Properties
The number of rows in the matrix. The column space and left null space live in Rm.
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Rows (m)
The number of columns in the matrix. The row space and null space live in Rn.
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Columns (n)
The rank is the number of linearly independent rows (or columns). It equals the dimension of both the column space and row space.
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Rank
The nullity is the dimension of the null space — the number of free variables, or "degrees of freedom" in solutions to Ax = 0. By the rank-nullity theorem: rank + nullity = n.
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Nullity
The determinant measures the signed volume scaling factor of the linear transformation. det = 0 means the matrix is singular (not invertible).
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Determinant
The trace is the sum of the diagonal entries. It also equals the sum of all eigenvalues.
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Trace
Eigenvalues are scalars λ satisfying Av = λv. They reveal stretching factors along principal axes. Complex eigenvalues indicate rotation.
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Eigenvalues
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The Four Subspaces
Column Space C(A)
All possible outputs Ax
Ambient: R^3
Null Space N(A)
All solutions to Ax = 0
Ambient: R^3
Row Space C(AT)
All combinations of rows
Ambient: R^3
Left Null Space N(AT)
All solutions to ATy = 0
Ambient: R^3
Key Relationships
C(A) ⊥ N(AT) — Orthogonal complements in Rm
C(AT) ⊥ N(A) — Orthogonal complements in Rn