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A subspace in linear algebra is a set of vectors that forms a smaller vector space within a larger vector space. It must satisfy three conditions: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication. These properties ensure that any linear combination of vectors within a subspace also belongs to that subspace. In our exercise, the subspace is spanned by two orthogonal vectors, \(\mathbf{u}_1\) and \(\mathbf{u}_2\). This means any vector in the subspace can be expressed as a linear combination of these vectors. Understanding subspaces helps describe solutions to linear equations and gives insight into vector spaces' structure. By projecting a vector onto a subspace, we can find a vector within the subspace that is closest to the original vector.

Orthogonal vectors are vectors that are perpendicular to each other. In mathematical terms, two vectors are orthogonal if their dot product is zero. In the context of the exercise, \(\mathbf{u}_1\) and \(\mathbf{u}_2\) are orthogonal vectors. This property greatly simplifies finding the orthogonal projection of a vector onto a subspace. When dealing with orthogonal vectors, the projections can be computed separately for each vector in the basis of the subspace. This separation is useful because it allows solving complex problems in steps. The orthogonality ensures that the components of the projection do not interfere with each other when added together.

The dot product is a way to multiply two vectors, resulting in a scalar. This operation is crucial in calculating projections. For vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as the sum of the products of their corresponding components. It can be represented as:

• \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \]

The dot product is significant because it measures how much one vector extends in the direction of another. In our exercise, the dot product is used to calculate the orthogonal projections onto \(\mathbf{u}_1\) and \(\mathbf{u}_2\).

• Calculate the dot product between \(\mathbf{v}\) and \(\mathbf{u}_1\)
• Calculate the dot product between \(\mathbf{v}\) and \(\mathbf{u}_2\)

These calculations help determine how much of \(\mathbf{v}\) is in the direction of each orthogonal basis vector.

Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces. It involves studying lines, planes, and subspaces and is applicable in numerous fields such as physics, engineering, and computer science. In this exercise, the concepts of vector spaces, orthogonality, and projections are all essential components of linear algebra.

• Vectors and operations on vectors, like addition and scalar multiplication, are fundamental.
• Subspaces like the one spanned by \(\mathbf{u}_1\) and \(\mathbf{u}_2\) fit within the larger vector space paradigm.
• Understanding orthogonal projections aids in solving many practical problems, such as finding the closest point in a subspace to a given vector.

Linear algebra provides tools and methodologies, like finding orthogonal projections, that are crucial in solving complex problems involving multiple dimensions.