Consider two subspaces V and W of ℝ^n . Let V+W be the set of all vectors in ℝ^n of the form v⃗+

step 1 For this exercise, we have the following. Vector K, vector L is an element of W plus V. We have

Consider two subspaces V and W of ℝ^n . Let V+W be the set...

Consider two subspaces $V$ and $W$ of $\mathbb{R}^{n} .$ Let $V+W$ be the set of all vectors in $\mathbb{R}^{n}$ of the form $\vec{v}+\vec{w},$ where $\vec{v}$ is in $V$ and $\vec{w}$ in $W$. Is $V+W$ necessarily a subspace of $\mathbb{R}^{n} ?$ If $V$ and $W$ are two distinct lines in $\mathbb{R}^{3},$ what is $V+W ?$ Draw a sketch.

Consider two subspaces $V$ and $W$ of $\mathbb{R}^{n} .$ Let $V+W$ be the set of all vectors in $\mathbb{R}^{n}$ of the form $\vec{v}+\vec{w},$ where $\vec{v}$ is in $V$ and $\vec{w}$ in $W$. Is $V+W$ necessarily a subspace of $\mathbb{R}^{n} ?$
If $V$ and $W$ are two distinct lines in $\mathbb{R}^{3},$ what is $V+W ?$ Draw a sketch.

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Key Concepts

Subspace

A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. This means it includes the zero vector, and any linear combination of vectors in the subspace remains in the subspace, ensuring the set itself behaves like a vector space.

Sum of Subspaces

The sum of two subspaces V and W, denoted V + W, is the set of all vectors that can be written as v + w where v is from V and w is from W. Since both V and W are subspaces, their sum retains closure under addition and scalar multiplication, making V + W a subspace of the original vector space.

Linear Independence and Span

When considering subspaces such as lines through the origin in ?3, if they are not collinear (i.e., they do not lie on the same line), any two nonzero vectors, one from each line, are linearly independent. Their linear combinations span a plane, forming a two-dimensional subspace of ?3.

Geometric Interpretation

Visualizing the sum of subspaces helps in understanding their structure. For instance, the sum of two distinct lines through the origin in ?3 results in a plane that contains both lines, with the intersection at the origin. Such geometric sketches assist in comprehending the behavior of vector addition in different dimensions.

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