Suppose U and V are subspaces of Rn. Prove that (U + V ) = U V

Which of the following are subspaces? Justify your answer in each case. a. \(\left\{\mathbf{x} \in \mathbb{R}^{2}: x_{1}+x_{2}=1\right\}\) b. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: \mathbf{x}=\left[\begin{array}{c} a \\ b \\ a+b \end{array}\right] \text { for some } a, b \in \mathbb{R}\right\}\) c. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}+2 x_{2}<0\right\}\) d. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\right\}\) e. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0\right\}\) f. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1\right\}\) g. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: \mathbf{x}=s\left[\begin{array}{l} 2 \\ 1 \\ 1 \end{array}\right]+t\left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right] \text { for some } s, t \in \mathbb{R}\right\}\) h. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: \mathbf{x}=\left[\begin{array}{l} 3 \\ 0 \\ 1 \end{array}\right]+s\left[\begin{array}{l} 2 \\ 1 \\ 1 \end{array}\right]+t\left[\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right] \text { for some } s, t \in \mathbb{R}\right\}\) i. \(\left\{\mathbf{x} \in \mathbb{R}^{3}: \mathbf{x}=\left[\begin{array}{r} 2 \\ 4 \\ -1 \end{array}\right]+s\left[\begin{array}{l} 2 \\ 1 \\ 1 \end{array}\right]+t\left[\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right] \text { for some } s, t \in \mathbb{R}\right\}\)

Decide whether each of the following collections of vectors spans \(\mathbb{R}^3\). a. (1, 1, 1), (1, 2, 2) b. (1, 1, 1), (1, 2, 2), (1, 3, 3) c. (1, 0, 1), (1, ?1, 1), (3, 5, 3), (2, 3, 2) d. (1, 0, ?1), (2, 1, 1), (0, 1, 5)

Criticize the following argument: For any vector v, we have 0v = 0. So the first criterion for subspaces is, in fact, a consequence of the second criterion and could therefore be omitted.

Let A be an \(n \times n\) matrix. Verify that \(V = \{\mathbf x \in \ \mathbb{R}^n : A \mathbf {x} = 3 \mathbf {x}\}\) is a subspace of \(\mathbb{R}^n\).

Let A and B be \(m \times n\) matrices. Show that \(V=\left\{\mathbf{x} \in \mathbb{R}^{n}: A \mathbf{x}=B \mathbf{x}\right\}\) is a subspace of \(\mathbb{R}^{n}\).

a. Let U and V be subspaces of \(\mathbb{R}^n\). Define the intersection of U and V to be \(U \cap V=\left\{\mathbf{x} \in \mathbb{R}^n: \mathbf{x} \in U\right.\) and \(\left.\mathbf{x} \in V\right\}\). Show that \(U \cap V\) is a subspace of \(\mathbb{R}^n\). Give two examples. b. Is \(U \cup V=\left\{\mathbf{x} \in \mathbb{R}^n: \mathbf{x} \in U\right.\) or \(\left.\mathbf{x} \in V\right\}\) always a subspace of \(\mathbb{R}^n\) ? Give a proof or counterexample.

Prove that if U and V are subspaces of Rn andW is a subspace of Rn containing all the vectors of U and all the vectors of V (that is, U W and V W), then U + V W. This means that U + V is the smallest subspace containing both U and V .

Let \(\mathbf{v}_1, \ldots, \mathbf{v}_k \in \mathbb{R}^n\) and let \(\mathbf{v} \in \mathbb{R}^n\). Prove that \(\operatorname{Span}\left(\mathbf{v}_1, \ldots, \mathbf{v}_k\right)=\operatorname{Span}\left(\mathbf{v}_1, \ldots, \mathbf{v}_k, \mathbf{v}\right)\) if and only if \(\mathbf{v} \in \operatorname{Span}\left(\mathbf{v}_1, \ldots, \mathbf{v}_k\right)\).

Determine the intersection of the subspaces \(\mathcal{P}_{1} \text { and } \mathcal{P}_{2}\) in each case: a. \(\mathcal{P}_{1}=\operatorname{Span}((1,0,1),(2,1,2)), \mathcal{P}_{2}=\operatorname{Span}((1,-1,0),(1,3,2))\) b. \(\mathcal{P}_{1}=\operatorname{Span}((1,2,2),(0,1,1)), \mathcal{P}_{2}=\operatorname{Span}((2,1,1),(1,0,0))\) c. \(\mathcal{P}_{1}=\operatorname{Span}((1,0,-1),(1,2,3)), \mathcal{P}_{2}=\left\{\mathbf{x}: x_{1}-x_{2}+x_{3}=0\right\}\) d. \(\mathcal{P}_{1}=\operatorname{Span}((1,1,0,1),(0,1,1,0)), \mathcal{P}_{2}=\operatorname{Span}((0,0,1,1),(1,1,0,0))\) e. \(\mathcal{P}_{1}=\operatorname{Span}((1,0,1,2),(0,1,0,-1)), \mathcal{P}_{2}=\operatorname{Span}((1,1,2,1),(1,1,0,1))\)

Let \(V \subset \mathbb{R}^n\) be a subspace. Show that \(V \cap V^{\perp}=\{0\}\).

Suppose V and W are orthogonal subspaces of Rn, i.e., v w = 0 for every v V and every w W. Prove that V W = {0}. _

Suppose V and W are orthogonal subspaces of \(\mathbb{R}^n\), i.e., \(\mathbf{v} \cdot \mathbf{w}=0\) for every \(\mathbf{v} \in V\) and every \(\boldsymbol{w} \in W\). Prove that \(V \subset W^{\perp}\).

Let \(V \subset \mathbb{R}^n\) be a subspace. Show that \(V \subset\left(V^{\perp}\right)^{\perp}\). Do you think more is true?

Let V and W be subspaces of \(\mathbb{R}^{n}\) with the property that \(V \subset W\). Prove that \(W^{\perp} \subset V^{\perp}\).

Let A be an \(m \times n\) matrix. Let \(V \subset \mathbb{R}^{n} \text { and } W \subset \mathbb{R}^{m}\) be subspaces. a. Show that \(\left\{\mathbf{x} \in \mathbb{R}^{n}: A \mathbf{x} \in W\right\}\) is a subspace of \(\mathbb{R}^{n}\). b. Show that \(\left\{\mathbf{y} \in \mathbb{R}^{m}: \mathbf{y}=A \mathbf{x} \text { for some } \mathbf{x} \in V\right\}\) is a subspace of \(\mathbb{R}^{m}\).

Suppose A is a symmetric n n matrix. Let V Rn be a subspace with the property that Ax V for every x V . Show that Ay V for all y V .

Use Exercises 13 and 14 to prove that for any subspace V Rn, we have V = _ (V ) _ .

Suppose U and V are subspaces of Rn. Prove that (U + V ) = U V .