91Ó°ÊÓ

In linear algebra, a **subspace** is a special subset of a vector space that, itself, is a vector space under the same operations as the larger space. To qualify as a subspace, a set must:

• Include the zero vector.
• Be closed under vector addition; that means, if you take any two vectors in the subspace, their sum should also be in the subspace.
• Be closed under scalar multiplication; if you multiply any vector by any scalar, the result should be in the subspace too.

For the given exercise, the subspaces are:

• **U**: This is the subspace of all vectors in the form \( (a, a, a) \), indicating that all three components must be equal.
• **W**: This includes vectors of the form \( (0, b, c) \), which lie on the yz-plane, showcasing that the x-component is always zero.

Understanding these conditions is crucial because it allows us to define spaces within larger spaces and thus solve many different types of linear equations.

Vector decomposition involves breaking down a vector into two or more vectors whose sum equals the original vector. When talking about the **direct sum** of subspaces as in \( R^3 = U \oplus W \), every vector in the space can be uniquely written as a sum of vectors from each subspace.
This can be a very powerful concept in linear algebra, enabling easier calculations and deeper insights.In this exercise:

• Any vector \( v = (x, y, z) \) in \( R^3 \) can be decomposed uniquely into vectors \( u \) and \( w \), such that:
• Vector \( u = (a, a, a) \) belongs to subspace \( U \).
• Vector \( w = (0, b, c) \) belongs to subspace \( W \).

One main aspect of any decomposition is uniqueness, ensuring that no other combinations of values for \( a \), \( b \), and \( c \) will satisfy the conditions for another set \( v = u + w \). Our calculations reveal how specific operations within these rules create that unique decomposition for any vector in the specified space.

**Linear algebra** is a branch of mathematics that deals with vectors, vector spaces, and linear transformations between these spaces. It forms the basis of much of modern applied mathematics and is used extensively in science and engineering.Key concepts of linear algebra include:

• **Vector Spaces:** Collections of vectors that can be added together and multiplied by scalars to get another vector in the same space.
• **Subspaces:** Special vector spaces contained within larger vector spaces, as defined earlier.
• **Linear Transformations:** Functions that preserve vector addition and scalar multiplication, mapping between different vector spaces.
• **Direct Sums:** A way to combine subspaces in such a manner that each element of the resultant space is uniquely decomposable into elements of the original subspaces.

This exercise, involving the direct sum of two subspaces of \( R^3 \), illustrates how we apply these principles practically: it involves synthesis, problem-solving, and a methodical approach—all of which are foundational in every linear algebra application.