Orthogonal projection

Task number: 2712

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

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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