Question

What is the dimension of R^n over R. write it in vectors.?

Answer 1

Answer on Question #40118, Math, Linear Algebra

What is the dimension of $R^n$ over $R$? Write it in vectors?

Answer

Definition: If a vector space $V$ has a basis consisting of $n$ vectors, then we say that dimension of $V$ is $n$. We also write $\dim(V) = n$.

$$\dim(\mathbb{R}^n) = n.$$

This is because the standard basis

$$\overline{e_1} = (1, 0, 0, \dots, 0), \overline{e_2} = (0, 1, 0, \dots, 0), \dots, \overline{e_n} = (0, 0, 0, \dots, 1)$$

consist of $n$ elements.

Also

$$\dim(\mathbb{R}^n) = \operatorname{tr}(id_{\mathbb{R}^n}) = \operatorname{tr} \begin{pmatrix} 1 & \dots & 0 \\ \vdots & 1 & \vdots \\ 0 & \dots & 1 \end{pmatrix} = n.$$