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Question

Find the orthogonal projection of $x = \begin{bmatrix}7 \\ 0 \\ -4 \\ -4 \end{bmatrix}$ onto the subspace of $\mathbb R^4$ spanned by $v_1 = \begin{bmatrix}-4 \\ 2 \\ 2 \\ -4 \end{bmatrix}, v_2 =\begin{bmatrix}2 \\ 2 \\ 2 \\ 0 \end{bmatrix}$

Answer

So I projected $x$ onto $v_1$ and $v_2$ and got $x_{v1} = \begin{bmatrix}2 \\ -1 \\ -1 \\ 2 \end{bmatrix} and, x_{v2} = \begin{bmatrix}1 \\ 1 \\ 1 \\ 0 \end{bmatrix}$ respectively.

I then subtracted $x_{v1}$ and $x_{v2}$ from $x$ and got $\begin{bmatrix}4 \\ 0 \\ -4 \\ -6 \end{bmatrix}$

which is wrong, and the correct answer should be $\begin{bmatrix}3 \\ 0 \\ 0 \\ 2 \end{bmatrix}$

So what have I done wrong?

  • 0
    @DylanMoreland Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3$1$38). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9$1$41) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2015-05-05

0 Answers 0