---
tag: 0R84PF4JV7
kind: statement
role: corollary
name: Dimension of a tensor product
msc: [15A69, 15A03]
uses: [0SH5VT8QYA, 05G2G0P224]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_product
    locator: Dimension
---

## Statement

Let $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then
$$\dim (V \otimes W) = \dim V \cdot \dim W,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]] over $F$.

## Remarks

It follows from the [[basis of a tensor product|0SH5VT8QYA]], which has $\dim V \cdot \dim W$ vectors.
For example, $F^m \otimes F^n$ has dimension $mn$, like the space of $m \times n$ matrices.
Compare this with the direct sum: $\dim (V \oplus W) = \dim V + \dim W$.

## Explanations

### L1 human

The dimension of a tensor product is the product of the dimensions.
The dimension of a direct sum is their sum instead.
So $\R^2 \otimes \R^3$ has dimension $6$.