91Ó°ÊÓ

Let \(U\) and \(W\) denote subspaces of \(\mathbb{R}^{n}\) and assume that \(U \subseteq W\). If \(\operatorname{dim} W=1\), show that either \(U=\\{\boldsymbol{0}\\}\) or \(U=W\)

We often write vectors in \(\mathbb{R}^{n}\) as rows. If \(S\) and \(T\) are nonempty sets of vectors in \(\mathbb{R}^{n}\), and if \(S \subseteq T\), show that \(\operatorname{span}\\{S\\} \subseteq \operatorname{span}\\{T\\}\).

We often write vectors in \(\mathbb{R}^{n}\) as rows. Let \(U\) and \(W\) be subspaces of \(\mathbb{R}^{n}\). Define their intersection \(U \cap W\) and their sum \(U+W\) as follows: $$ U \cap W=\left\\{\mathbf{x} \in \mathbb{R}^{n} \mid \mathbf{x} \text { belongs to both } U \text { and } W\right\\} $$ \(U+W=\left\\{\mathbf{x} \in \mathbb{R}^{n} \mid \mathbf{x}\right.\) is a sum of a vector in \(U\) and a vector in \(W\\}\). a. Show that \(U \cap W\) is a subspace of \(\mathbb{R}^{n}\). b. Show that \(U+W\) is a subspace of \(\mathbb{R}^{n}\).

We often write vectors in \(\mathbb{R}^{n}\) as rows. Let \(P\) denote an invertible \(n \times n\) matrix. If \(\lambda\) is a number, show that $$ E_{\lambda}\left(P A P^{-1}\right)=\left\\{P \mathbf{x} \mid \mathbf{x} \text { is in } E_{\lambda}(A)\right\\} $$ for each \(n \times n\) matrix \(A\).

We often write vectors in \(\mathbb{R}^{n}\) as rows. Show that every proper subspace \(U\) of \(\mathbb{R}^{2}\) is a line through the origin. [Hint: If \(\mathbf{d}\) is a nonzero vector in \(U,\) let \(L=\mathbb{R} \mathbf{d}=\\{r \mathbf{d} \mid r\) in \(\mathbb{R}\\}\) denote the line with direction vector \(\mathbf{d}\). If \(\mathbf{u}\) is in \(U\) but not in \(L\), argue geometrically that every vector \(\mathbf{v}\) in \(\mathbb{R}^{2}\) is a linear combination of \(\mathbf{u}\) and \(\mathbf{d}\).