## Orthogonal projection

For matrix from the previous exercise calculate the orthogonal projection $$\mathbf p$$ of the vector $$\mathbf a=(2, 2, 1, 5)^T$$ to the row space of the matrix. Also determine coordinates $$[\mathbf p]_Z$$ w.r.t. the basis $$Z$$.

• #### Variant 1

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

• #### Variant 2

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

• #### Variant 3

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

• #### Variant 4

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