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Vector projection helps us find how much of a vector points in the direction of another vector. Imagine you're casting a shadow of one vector onto another. That's exactly what a vector projection is! This technique is especially useful when dealing with subspaces, which in this exercise, is represented by vector \( \mathbf{w} \).
When projecting vector \( \mathbf{v} \) onto the span of vector \( \mathbf{w} \), we use the formula:

• First, calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \), which measures how much \( \mathbf{v} \) aligns with \( \mathbf{w} \).
• Then, divide by the dot product \( \mathbf{w} \cdot \mathbf{w} \), essentially finding the length squared of \( \mathbf{w} \).
• Finally, multiply by \( \mathbf{w} \) to get the actual projection vector.

This projection represents the component of \( \mathbf{v} \) that lies in the direction defined by \( W \). It's like saying, "this much of \( \mathbf{v} \) is aligned with \( W \)."In this exercise, we found this projection to be \( \left[ \frac{4}{3}, \frac{4}{3}, \frac{4}{3} \right] \), showing exactly how \( \mathbf{v} \) spreads itself along \( \mathbf{w} \).

Orthogonal components help in decomposing a vector into parts that do not interfere with each other in terms of direction. In geometric terms, orthogonal means two vectors meet at a right angle, just like the edges of a square corner.
After projecting \( \mathbf{v} \) onto the subspace \( W \), there remains a part of \( \mathbf{v} \) that doesn't align with this subspace. This remaining vector is the orthogonal component of \( \mathbf{v} \). In mathematical terms, this is calculated by subtracting the projection of \( \mathbf{v} \) from itself:

• Orthogonal Component \( \mathbf{v}_{\perp} = \mathbf{v} - \operatorname{proj}_{W}(\mathbf{v}) \)

This component ensures that \( \mathbf{v} \) is perfectly split into parts lying inside \( W \) and those outside it. In this exercise, the orthogonal component was calculated as \( \left[ \frac{5}{3}, \frac{2}{3}, -\frac{7}{3} \right] \), representing the remainder of \( \mathbf{v} \) not in \( W \).
The nicer part? This orthogonal component is completely separate in terms of direction from \( W \), meaning there's no overlap between them, just like independent directions on a compass.

A linear subspace is a collection of vectors that maintain operation under vector addition and scalar multiplication. It's like a specific "path" or "plane" within a larger space defined by a set of vectors. In this exercise, \( W \) is a subspace spanned by the vector \( \mathbf{w} = \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} \).
Think of \( W \) as all possible directions you could get by multiplying \( \mathbf{w} \) by any scalar and adding them. It's like drawing a straight line in space, but our line has infinite length because of these scalars.
Key properties of a linear subspace include:

• Every linear combination of vectors in the subspace is still within the subspace.
• The subspace must include the zero vector.

By projecting \( \mathbf{v} \) onto \( W \), we essentially find the part of \( \mathbf{v} \) that "lives" on this straight line, which is critical when we want to understand how vectors interact within given constraints. This concept is pivotal in ensuring that while we break down a vector like \( \mathbf{v} \), we do it predictably and within what's defined by our starting subspace.