91Ó°ÊÓ

Vector spaces form the foundation of linear algebra, providing a comprehensive structure with defined operations of addition and scalar multiplication. These operations adhere to specific rules and produce outputs, or vectors, within the same vector space. This makes vector spaces a keystone in understanding higher-dimensional algebraic systems. A vector space, at its core, is a collection of vectors. It is equipped with two operations:

• Addition: Combining two vectors results in another vector within the same space.
• Scalar multiplication: Stretching or shrinking a vector by a scalar, leading to another member of the same space.

For example, the vector \(\mathbf{y}=\begin{bmatrix}4 \ 8 \ 1\end{bmatrix}\) is part of a three-dimensional vector space. This means if any scalar is used to multiply this vector or if it is added to another vector in the same space, the result also lies within this space.
In our example, the vectors \(\mathbf{u}_1\) and \(\mathbf{u}_2\) form a basis for the subspace spanned by them, denoted as \(W= \operatorname{Span}\{\mathbf{u}_1, \mathbf{u}_2\}\). This subspace, \(W\), is contained within the original vector space and itself possesses the vector space properties.

Matrix multiplication is a cornerstone operation essential for various transformations in linear algebra. Specifically, it relates to how matrices affect vectors and other matrices. The core idea is to take the input matrix's rows and columns and combine them in a way that respects linear transformations.When multiplying two matrices, it's crucial to remember:

• The number of columns in the first matrix must match the number of rows in the second one.
• The resulting matrix takes its dimensions from the rows of the first and the columns of the second matrix.

Let's address the operations performed in the exercise. Calculate \(U^T U\), where \(U^T\) is the transpose of \(U\). The transpose flips the rows with columns of \(U\), and multiplying it with \(U\) creates a matrix that provides comprehensive information about the linear relationships between the vectors forming \(U\). This matrix multiplication reflects an identity matrix that conveniently showcases the orthonormality of the basis vectors— \[U^T U = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\].
In simpler terms, this operation shows us that vectors \(\mathbf{u}_1\) and \(\mathbf{u}_2\) are orthonormal, meaning they are at right angles to each other and each have a length of one.

Orthogonal projection is a method used to project a vector onto a subspace, which often retains the most essential components of the original vector within that specific subspace. This is particularly useful in reducing dimensions while preserving crucial information.Orthogonal projection is defined mathematically by the formula:
\[ \text{proj}_W \mathbf{y} = U(U^T U)^{-1} U^T \mathbf{y} \]
where \(U\) comprises the basis of the subspace to which the vector is projected.
In this exercise, to find the orthogonal projection of \(\mathbf{y}\) onto the space \(W\), we leverage the fact that \(U^T U = I\), simplifying the formula to:
\[ \text{proj}_W \mathbf{y} = U U^T \mathbf{y} \]The calculation shows that the projecting results transform the vector into a new vector form \((-\frac{2}{3}, \frac{17}{3}, \frac{16}{3})\), indicating how much of \(\mathbf{y}\) aligns with span \(W\). This is confirmed by performing the same transformation using the matrix product \( (U U^T) \mathbf{y} \), where \(U U^T\) acts as the projection matrix.