Linear Transformation Calculator
Linear transformations and vector spaces with exact fractions: the matrix of T relative to any bases, coordinates and change of basis, kernel and range with rank–nullity, the subspace test, and span membership.
About the Linear Transformation Calculator
A linear map T: Rⁿ → Rᵐ is fixed by its matrix. Relative to a basis B of the domain and C of the target, column j of [T] is the C-coordinates of T(bⱼ), so [T] = C⁻¹·T(B). Coordinates of v in a basis B solve c₁b₁ + … + cₙbₙ = v, and P = C⁻¹B changes B-coordinates to C-coordinates.
The kernel (vectors sent to 0) and the range have dimensions nullity and rank that add to n. A subset W is a subspace when it contains 0 and is closed under addition and scalar multiples; the solution set of linear equations equal to 0 always is.
The formula
[T]_B^C = C⁻¹ · T(B); rank + nullity = n
Worked example
T(x, y, z) = (x + y, y − z)
- Kernel: x + y = 0, y = z, so (−1, 1, 1)
- rank 2 + nullity 1 = 3
Questions people ask
Why does a linear map send 0 to 0?
T(0) = T(0·v) = 0·T(v) = 0. So T(x) = x + 1 isn't linear.
Is a line through the origin a subspace?
Yes. A line or plane not through the origin isn't, because it misses 0.