diff --git a/nitransforms/base.py b/nitransforms/base.py
index 68b97f75..96f00edb 100644
--- a/nitransforms/base.py
+++ b/nitransforms/base.py
@@ -178,7 +178,7 @@ def __ne__(self, other):
 class TransformBase:
     """Abstract image class to represent transforms."""
 
-    __slots__ = ("_reference",)
+    __slots__ = ("_reference", "_ndim",)
 
     def __init__(self, reference=None):
         """Instantiate a transform."""
@@ -220,7 +220,7 @@ def reference(self, image):
     @property
     def ndim(self):
         """Access the dimensions of the reference space."""
-        return self.reference.ndim
+        raise TypeError("TransformBase has no dimensions")
 
     def apply(
         self,
diff --git a/nitransforms/linear.py b/nitransforms/linear.py
index eb4a95d7..af14f396 100644
--- a/nitransforms/linear.py
+++ b/nitransforms/linear.py
@@ -143,6 +143,11 @@ def matrix(self):
         """Access the internal representation of this affine."""
         return self._matrix
 
+    @property
+    def ndim(self):
+        """Access the internal representation of this affine."""
+        return self._matrix.ndim + 1
+
     def map(self, x, inverse=False):
         r"""
         Apply :math:`y = f(x)`.
diff --git a/nitransforms/tests/test_base.py b/nitransforms/tests/test_base.py
index 8422ca10..07a7e4ec 100644
--- a/nitransforms/tests/test_base.py
+++ b/nitransforms/tests/test_base.py
@@ -94,7 +94,8 @@ def _to_hdf5(klass, x5_root):
     # Test identity transform
     xfm = TransformBase()
     xfm.reference = fname
-    assert xfm.ndim == 3
+    with pytest.raises(TypeError):
+        _ = xfm.ndim
     moved = xfm.apply(fname, order=0)
     assert np.all(
         imgdata == np.asanyarray(moved.dataobj, dtype=moved.get_data_dtype())
@@ -103,12 +104,19 @@ def _to_hdf5(klass, x5_root):
     # Test identity transform - setting reference
     xfm = TransformBase()
     xfm.reference = fname
-    assert xfm.ndim == 3
+    with pytest.raises(TypeError):
+        _ = xfm.ndim
     moved = xfm.apply(str(fname), reference=fname, order=0)
     assert np.all(
         imgdata == np.asanyarray(moved.dataobj, dtype=moved.get_data_dtype())
     )
 
+    # Test ndim returned by affine
+    assert nitl.Affine().ndim == 3
+    assert nitl.LinearTransformsMapping(
+        [nitl.Affine(), nitl.Affine()]
+    ).ndim == 4
+
     # Test applying to Gifti
     gii = nb.gifti.GiftiImage(
         darrays=[