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A subspace is an important concept in linear algebra. It is a set of vectors that satisfy two main properties: any linear combination of vectors in the subspace remains in the subspace, and the zero vector is always included. To consider a set of vectors as forming a subspace, ensure the following:

• Make sure that any sum of two vectors from the set is also a vector in the subspace.
• The scalar multiplication of any vector in the set by a real number should also remain within the subspace.

In the original exercise, vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) span a subspace \( W \). This means any linear combination of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) will also be in the subspace \( W \), showing the set is indeed a subspace. Additionally, ensuring that \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) are linearly independent confirms there is no other dependency relation narrowing the dimension of this subspace beyond the vectors given.

Linear independence occurs when no vector in a set can be written as a linear combination of the others. This means each vector adds a new dimension of freedom. If vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \), in this exercise, were dependent, one could express one vector entirely in terms of the other.To check for independence:

• Determine if there are non-zero scalars \( a \) and \( b \) such that \( a\mathbf{u}_1 + b\mathbf{u}_2 = \mathbf{0} \).
• If the only solution is \( a = 0 \) and \( b = 0 \), the vectors are independent.

Here, these vectors don't form a linear combination ending in zero with non-trivial coefficients, which confirms their independence. Determining linear independence is crucial for understanding the span and dimension of vector spaces.

A linear combination involves creating a new vector by adding together scalar multiples of other vectors. For instance, in this exercise, we wish to express the vector \( \mathbf{y} \) as a sum of multiples of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) (the basis vectors of the subspace) and a vector \( \mathbf{z} \) orthogonal to the subspace.To express \( \mathbf{y} \) as a linear combination of the basis vectors of subspace \( W \):

• Use coefficients \( a \) and \( b \) determined by the projection calculations using dot products and the subspace basis' lengths.
• These coefficients then multiply \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) respectively.

Expressing any vector like \( \mathbf{y} \) in terms of a subspace's basis gives insight into ways these vectors can be manipulated to reach or project onto other locations within or adjacent to that space.

Orthogonality is a relationship between vectors denoting perpendicularity. Two vectors are orthogonal if their dot product is zero. This concept is critical in breaking down vectors into components parallel and perpendicular to a particular subspace, as done in the exercise.One key property in orthogonality involves the separation of vectors into projected (within the subspace) and orthogonal (outside the subspace) components:

• The projection of a vector \( \mathbf{y} \) onto a subspace \( W \) falls entirely within this subspace, calculated using dot products with basis vectors.
• The orthogonal component \( \mathbf{z} = \mathbf{y} - \mathbf{y}_W \) lies completely perpendicular to any vector within \( W \).

Checking the orthogonality of \( \mathbf{z} \) against \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) confirms accurate projection, which in applications, is important for minimizing error and understanding geometric relationships in spaces.