Question

Let \(U\) and \(W\) be subspaces of \(V\). a. Show that \(U+W\) is a subspace of \(V\) containing both \(U\) and \(W\). b. Show that \(\operatorname{span}\\{\mathbf{u}, \mathbf{w}\\}=\mathbb{R} \mathbf{u}+\mathbb{R} \mathbf{w}\) for any vectors \(\mathbf{u}\) and \(\mathbf{w}\). c. Show that $$\begin{array}{l}\operatorname{span}\left\\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{m}, \mathbf{w}_{1}, \ldots,\mathbf{w}_{n}\right\\} \\ =\operatorname{span}\left\\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{m}\right\\}+\operatorname{span}\left\\{\mathbf{w}_{1}, \ldots, \mathbf{w}_{n}\right\\} \end{array}$$ for any vectors \(\mathbf{u}_{i}\) in \(U\) and \(\mathbf{w}_{j}\) in \(W\).

Short answer

(a) Yes, \(U+W\) is a subspace. (b) \(\operatorname{span}\{\mathbf{u}, \mathbf{w}\} = \mathbb{R}\mathbf{u} + \mathbb{R}\mathbf{w}\). (c) The spans are equal as described.

Step-by-step solution

Define the Sum of Subspaces

Given vector spaces or subspaces \(U\) and \(W\) of a vector space \(V\), the sum \(U+W\) is defined as the set of all vectors that can be written as \(\mathbf{u} + \mathbf{w}\), where \(\mathbf{u} \in U\) and \(\mathbf{w} \in W\). We need to show that \(U+W\) is a subspace of \(V\).

Check Non-emptiness of U+W

Since \(U\) and \(W\) are subspaces, they contain the zero vector. Therefore, \(0 = 0_U + 0_W \in U+W\). So, \(U+W\) is non-empty.

Check Closure Under Addition for U+W

Take any \(\mathbf{a}, \mathbf{b} \in U+W\). Then \(\mathbf{a} = \mathbf{u}_1 + \mathbf{w}_1\) and \(\mathbf{b} = \mathbf{u}_2 + \mathbf{w}_2\), where \(\mathbf{u}_1, \mathbf{u}_2 \in U\) and \(\mathbf{w}_1, \mathbf{w}_2 \in W\). Then \(\mathbf{a} + \mathbf{b} = (\mathbf{u}_1 + \mathbf{u}_2) + (\mathbf{w}_1 + \mathbf{w}_2) \in U+W\) since both \(U\) and \(W\) are closed under addition. Thus, \(U+W\) is closed under addition.

Check Closure Under Scalar Multiplication for U+W

Take any \(\mathbf{a} \in U+W\) and any scalar \(c\). Then \(\mathbf{a} = \mathbf{u} + \mathbf{w}\), where \(\mathbf{u} \in U\) and \(\mathbf{w} \in W\). Then \(c\mathbf{a} = c(\mathbf{u} + \mathbf{w}) = c\mathbf{u} + c\mathbf{w} \in U+W\) because \(U\) and \(W\) are closed under scalar multiplication. Thus, \(U+W\) is closed under scalar multiplication.

Conclusion for Part (a)

Since \(U+W\) is non-empty, closed under addition, and closed under scalar multiplication, \(U+W\) is a subspace of \(V\) containing both \(U\) and \(W\).

Define the Span of Vectors

The span of vectors \(\{\mathbf{u}, \mathbf{w}\}\) is the set of all linear combinations \(a\mathbf{u} + b\mathbf{w}\), where \(a, b\) are scalars. This is denoted by \(\operatorname{span}\{\mathbf{u}, \mathbf{w}\}\).

Recognize Span as a Subspace

The span of \(\{\mathbf{u}, \mathbf{w}\}\) is \(\mathbb{R}\mathbf{u} + \mathbb{R}\mathbf{w}\), which can be shown to include all linear combinations from the individual spans of \(\mathbf{u}\) and \(\mathbf{w}\). Here, \(\mathbb{R}\mathbf{u}\) is the set of all multiples of \(\mathbf{u}\) and \(\mathbb{R}\mathbf{w}\) is the set of all multiples of \(\mathbf{w}\), which are also spans.

Conclusion for Part (b)

The span \(\operatorname{span}\{\mathbf{u}, \mathbf{w}\}\) is equivalent to \(\mathbb{R}\mathbf{u} + \mathbb{R}\mathbf{w}\) as they both contain exactly the same linear combinations of the vectors. Thus, they are equal.

Define General Span Combination

For vectors \(\mathbf{u}_i\) from \(U\) and \(\mathbf{w}_j\) from \(W\), their spans are \(\operatorname{span}\{\mathbf{u}_1, \ldots, \mathbf{u}_m\}\) and \(\operatorname{span}\{\mathbf{w}_1, \ldots, \mathbf{w}_n\}\) respectively.

Express Combined Span as a Sum

The span of \(\{\mathbf{u}_1, \ldots, \mathbf{u}_m, \mathbf{w}_1, \ldots, \mathbf{w}_n\}\) includes all vectors formed by linear combinations of both sets. This is the same as \(\operatorname{span}\{\mathbf{u}_1, \ldots, \mathbf{u}_m\} + \operatorname{span}\{\mathbf{w}_1, \ldots, \mathbf{w}_n\}\).

Conclusion for Part (c)

The equality holds because any linear combination of the full set of vectors can be decomposed into separate linear combinations of each subgroup, and vice versa.

Key concepts

Vector Spaces

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars. These operations follow certain rules or axioms, such as associativity, distributivity, and the existence of an identity element. In simple terms, a vector space allows you to scale and combine vectors while still remaining within the same type of space. Common examples of vector spaces include Cartesian coordinates in geometry, where vectors can represent points in a plane or space.

In a mathematical sense, considering a vector space helps to structure the solutions to various problems by providing a consistent framework. It's like setting the rules for how you can manipulate elements within the space, which can be tremendously helpful in understanding complex structures and systems.

Linear Combination

A linear combination involves taking a set of vectors and combining them via addition, scaled by coefficients or scalars. For instance, if you have vectors \( \f{u}\) and \( \f{w}\), a linear combination would be written as \( a \f{u} + b \f{w} \), where \( a \) and \( b \) are scalars.

This concept is crucial because it forms the groundwork for phenomena like the span of vectors or the components of a vector space. Through linear combinations, you can form new vectors, potentially exploring new dimensions within a space. By changing scalars, you can stretch or shrink vectors, illustrating a vector's influence and relation in space. This makes the concept a cornerstone in topics like linear algebra and geometry.

Span of Vectors

The span of a set of vectors is essentially all possible vectors you can reach by taking linear combinations of those vectors. If you have vectors \( \f{u}, \f{v}, \) and \( \f{w} \), their span is the set of all vectors \( a\bf{u} + b\bf{v} + c\bf{w} \), where \( a, b, \) and \( c \) are scalars.

Understanding the span is important because it helps determine the dimensions you can cover with a given set of vectors. A span can form subspaces within a larger vector space, providing crucial information about vector dependencies and the coverage of the vector space. This is fundamental in applications such as solving systems of linear equations and in transformations within geometric spaces.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, effectively scaling (stretching or shrinking) it while maintaining its direction. If a vector \( \f{v} \) is multiplied by a scalar \( c \), the result is \( c\bf{v} \), changing the vector's magnitude without altering its orientation.

In practice, scalar multiplication is vital because it enables the adjustment of a vector's size within a vector space while adhering to the rules of the space. For mathematical modeling and computations, scalar multiplication is a basic yet powerful tool for adjusting vector properties without losing the vector's fundamental direction or path. This operation underpins the principles of linear transformations and helps to link the physical representation of vectors with their algebraic expressions.