91Ó°ÊÓ

When dealing with vectors, the concept of orthogonal projection is crucial for understanding how a vector can be represented within a subspace. The orthogonal projection of a vector \(\mathbf{y}\) onto another vector \(\mathbf{u}\) is essentially the component of \(\mathbf{y}\) that points in the direction of \(\mathbf{u}\). This is especially useful in finding how much of \(\mathbf{y}\) can be described within the span of \(\mathbf{u}\) or a set of basis vectors.
To compute the orthogonal projection of \(\mathbf{y}\) onto a vector \(\mathbf{u}\), the formula is given by:

• \( \text{proj}_{\mathbf{u}}(\mathbf{y}) = \frac{\mathbf{y} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u} \)

This formula works because it scales the unit vector in the direction of \(\mathbf{u}\) by the amount of \(\mathbf{y}\) that is "projected" onto \(\mathbf{u}\). The result is a vector that lies directly along \(\mathbf{u}\).
An important aspect of orthogonal projection in multi-dimensional space, particularly with orthogonal sets, is ensuring that none of the original vector's full magnitude is lost in the process of breaking it down into components within the span of the vectors involved.

The dot product is a fundamental operation in vector algebra that helps determine the relationship between two vectors. It is a scalar value that gives insight into the angle between vectors and their respective magnitudes.
Calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 + \ldots \)

If the dot product of two vectors is zero, this signifies that the vectors are orthogonal, meaning they are at right angles to each other in the vector space.
For our example, checking the dot product of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) gives zero, confirming they are orthogonal. This orthogonality plays a key role in simplifying calculations for projections and for verifying the integrity of vector sets in higher-dimensional analysis. When working with an orthogonal set, each vector operates independently, minimizing complications when translating or synthesizing vector data.

The span of a set of vectors refers to all possible vectors that can be constructed as linear combinations of the set. This is an essential concept in understanding vector spaces because it defines the space created by the combination of multiple vectors.
In mathematical terms, given vectors \(\mathbf{u}_1\) and \(\mathbf{u}_2\), the span is defined as:

• \( \text{Span}\{\mathbf{u}_1, \mathbf{u}_2\} = \{c_1 \mathbf{u}_1 + c_2 \mathbf{u}_2 \mid c_1, c_2 \in \mathbb{R}\} \)

This means any vector in the space described by \(\mathbf{u}_1\) and \(\mathbf{u}_2\) can be expressed as a weighted sum of these vectors, where \(c_1\) and \(c_2\) are real numbers. In our exercise case, the span of \(\mathbf{u}_1\) and \(\mathbf{u}_2\) encapsulates a plane in the three-dimensional space.
Calculating the projection of a vector like \(\mathbf{y}\) onto this span thereby gives a clear representation of how much of \(\mathbf{y}\) can lie within this plane, effectively "projecting" it into the defined space. This is fundamental when solving systems of equations or optimizing functions, as it allows you to simplify complex three-dimensional problems into two dimensions.