Study matrix mappings, dimensions, rank, nullity, image, and kernel. Export clear results for linear algebra assignments and reports instantly today.
| Matrix Size | Linear Map | Domain | Codomain | Possible Rank Range | Meaning |
|---|---|---|---|---|---|
| 2 × 3 | T: R³ → R² | R³ | R² | 0 to 2 | Maps three inputs to two outputs. |
| 3 × 2 | T: R² → R³ | R² | R³ | 0 to 2 | Maps two inputs to three outputs. |
| 4 × 4 | T: R⁴ → R⁴ | R⁴ | R⁴ | 0 to 4 | May be invertible if rank is four. |
For an m × n matrix A, the associated linear transformation is:
T(x) = Ax
The input vector x must have n entries. So the domain is Rⁿ.
The output vector Ax has m entries. So the codomain is Rᵐ.
Rⁿ, where n is the number of columns.Rᵐ, where m is the number of rows.n − rank(A).rank(A) = n.rank(A) = m.A matrix can describe a linear transformation. This transformation takes an input vector and produces an output vector. The size of the matrix tells the allowed input size and output size. This is why the row and column count matters. An m by n matrix has m rows and n columns. It acts on vectors with n components. The result has m components. So the matrix maps R n into R m. This idea is central in linear algebra.
The domain is the set of valid input vectors. For a matrix, the domain dimension equals the number of columns. If a matrix has three columns, it can multiply a vector from R three. A vector with two or four entries will not fit. The calculator checks this structure automatically. It also shows the input space using clear notation.
The codomain is the target space for all possible outputs. Its dimension equals the number of rows. A matrix with two rows always produces a two-entry output vector. The actual outputs may not fill the whole codomain. That smaller set is called the image or column space. Rank measures the dimension of this image.
Rank and nullity add deeper meaning. Rank shows how many independent output directions exist. Nullity shows how many independent input directions collapse to zero. The rank-nullity theorem links both values. It says that rank plus nullity equals the domain dimension. This calculator applies that rule after reducing the matrix.
A matrix is injective when different inputs always give different outputs. This happens when rank equals the domain dimension. A matrix is surjective when its image fills the codomain. This happens when rank equals the codomain dimension. If both conditions hold, the transformation is bijective. Square full-rank matrices have this special property.
The domain is the input vector space. For an m by n matrix, the domain is R n because the matrix needs vectors with n entries.
The codomain is the target output space. For an m by n matrix, the codomain is R m because every product has m entries.
No. The codomain is the full target space. The image is the set of outputs the matrix actually reaches.
Rank gives the image dimension. It also helps decide whether the matrix transformation is injective, surjective, or bijective.
Nullity is the dimension of the kernel. It counts independent input directions that are mapped to the zero vector.
A matrix transformation is injective when its rank equals the number of columns. Then no nonzero input maps to zero.
It is surjective when its rank equals the number of rows. Then the image fills the entire codomain.
Yes. It supports rectangular and square matrices. Rows define codomain dimension, and columns define domain dimension.
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