## Orthonormal bases

### Task number: 2710

In $$\mathbb R^4$$ with the standard scalar product $$\langle \mathbf x|\mathbf y \rangle=\sum\limits_{i=1}^4 x_iy_i$$ determine by the Gram-Schmidt method an orthonormal basis $$Z=\{\mathbf z_1,…,\mathbf z_r\}$$ of the row space of the following matrix.

• #### Variant 1

$$\begin{pmatrix} 1 & 1 & 1 & 1 \\ 4 & 1 & 4 & 1 \\ 1 & 2 & 3 & 4 \\ \end{pmatrix}$$

• #### Variant 2

$$\begin{pmatrix} 0 & 3 & 4 & 0 \\ 0 & 0 & 5 & 0 \\ 2 & 1 & 0 & 2 \\ \end{pmatrix}$$

• #### Variant 3

$$\begin{pmatrix} 2 & 0 & 1 & 2 \\ 4 & 3 & 2 & 4 \\ 6 & -5 & 3 & 6 \\ -4 & 2 & 4 & 2 \\ \end{pmatrix}$$

• #### Variant 4

$$\begin{pmatrix} 2 & 4 & 2 & 1 \\ -1 & -2 & -2 & -1 \\ 1 & 2 & 4 & 2 \\ 1 & 2 & 3 & 4 \\ \end{pmatrix}$$