Question

How can I find the projection of a vector onto a subspace?

Answer 1

From the question given, we have been asked to find the projection of a vector onto a subspace. We can find the projection of a vector onto a subspace by using the concept of linear combination of vectors.

Complete step-by-step solution:
Projection of a vector onto a subspace:
First of all, for our convenience, let us assume a subspace.
Let us assume as a subspace of $S\subset V$
The subspace $S\subset V$ admits ${{u}_{1}},{{u}_{2}},..........{{u}_{n}}$ as an orthogonal basis.
This means that every vector $u\in S$ can be written as a linear combination of the ${{u}_{i}}$ vectors.
Representation of the vector as a linear combination of the ${{u}_{i}}$ vectors: $u=\sum\limits_{i=1}^{n}{{{a}_{i}}{{u}_{i}}}$
Now, assume that you want to project a certain vector $v\in V$ onto $S$. Of course, if in particular $v\in S$, then its projection is $v$ itself. Let us assume that $v\in V$ but $v\notin S$.
Let us call $u$ as the projection of $v$ onto $S$.
Following what we wrote before, we need to find the coefficients ${{a}_{i}}$ to express $u$ inside $S$.
Now, as discussed above the coefficients are ${{a}_{i}}=\dfrac{\left\langle v,{{u}_{i}} \right\rangle }{\left\langle {{u}_{i}},{{u}_{i}} \right\rangle }$
Therefore, we got the coefficients.
Now, we have to find the projection of the vector onto a subspace.
The projection of vector onto a subspace using the coefficient we found is $u=\sum\limits_{i=1}^{n}{\dfrac{\left\langle v,{{u}_{i}} \right\rangle }{\left\langle {{u}_{i}},{{u}_{i}} \right\rangle }}$
Hence, the projection of the vector onto a subspace is found.

Note: We should be well aware of the vectors concept. Also, we should be well aware of the properties of vectors. Also, we should be very careful while finding the coefficient. Also, we should make the assumptions very clear to get the correct answer for the given question. Also we should not get confused while doing the vector calculation.