Chapter 3 Exercise 02 (Finite-Dimensional Vector Spaces)

3.1 Exercise 2A

3.2 Exercise 2B (Bases)

  1. Find all vector spaces that have exactly one basis.

  2. Verify all assertions in Example

    1. Let \(W\) be the subspace of \(\mathbb{R}^5\) defined by \[W = \{ (x_1, x_2, x_3, x_4, x_5) \in \mathbb{R}^5 \mid x_1 = 3x_2 \text{ and } x_3 = 7x_4 \}. \] Find a basis of \(W\).

    2. Extend the basis in (a) to a basis of \(\mathbb{R}^5\).

    3. Find a subspace\(V\)of\(\mathbb{R}^5\)such that \(\mathbb{R}^5 = W \oplus V\).

Answer: a. Let \((x_1,x_2,x_3,x_4,x_5)\in U\). Then \(x_1=3x_2\) and \(x_3=7x_4\). Thus, \[\begin{align} (x_1,x_2,x_3,x_4,x_5) &= (3x_2,x_2,7x_4,x_4,x_5)\\ &=x_2(3,1,0,0,0)+x_4(0,0,7,1,0)+(0,0,0,0,1) \end{align}\]

Let \(v_1:=(3,1,0,0,0),v_2:=(0,0,7,1,0)\) and \(v_3:=(0,0,0,0,1)\). Since \(x_2,x_4,x_5\in \mathbb{R}\),then

\[ \text{span}(v_1,v_2,v_3)=U \] Now, we need to show that \(v_1,v_2,v_3\) are linearly independent.Suppose that there exist \(a_1,a_2,a_3\in \mathbb{R}\) such that, \[\begin{align} a_1(3,1,0,0,0)+a_2(0,0,7,1,0)+a_3(0,0,0,0,1) &=(0,0,0,0,0)\\ (3a_1,a_1,7a_2,a_2,a_3)&=(0,0,0,0,0). \end{align}\] Thus, \(a_1=a_2=a_3=0\). Hence, \(v_1,v_2,v_3\) are linearly independent.

Therefore, the list \(v_1,v_2,v_3\) is a basis of \(U\).

  1. Let \(v_4:=(0,1,0,0,0)\) and \(v_5:=(0,0,1,0,0)\).

Claim 1: \(\text{span}(v_1,v_2,v_3,v_4,v_5)=\mathbb{R}^5\).

Let \((y_1,y_2,y_3,y_4,y_5)\in \mathbb{R}^5\). Then, \[\begin{align} (y_1,y_2,y_3,y_4,y_5) &= (y_1,y_2+\frac{y_1}{3}-\frac{y_1}{3},y_3+7y_4-7y_4,y_4,y_5)\\ &=\frac{y_1}{3}(3,1,0,0,0)+y_4(0,0,7,1,0)+ y_5(0,0,0,0,1)+\left(y_2-\frac{y_1}{3}\right)(0,1,0,0,0)+\left(y_3-7y_4\right) \end{align}\]

Since \(y_1,...,y_5\in \mathbb{R}\), we have established that the list \((3, 1, 0, 0, 0), (0, 0, 7, 1, 0), (0, 0, 0, 0, 1),(0,1,0,0,0),(0,0,1,0,0)\) spans \(\mathbb{R}^5\)

Claim 2: the list of vectors \(v_1,v_2,v_3,v_4,v_5\) are linearly independent.

Suppose that there exist \(b_1,...,b_5\in \mathbb{R}\) such that \[\begin{align} b_1v_1+b_2v_2+b_3v_3+b_4v_4+b_5v_5 &=0\\ b_1(3, 1, 0, 0, 0) + b_2(0, 0, 7, 1, 0) + b_3(0, 0, 0, 0, 1) + b_4(0, 1, 0, 0, 0) + b_5(0, 0, 1, 0, 0) &= (0, 0, 0, 0, 0)\\ (3b1, b1 + b4, 7b2 + b5, b2, b3) &= (0, 0, 0, 0, 0). \end{align}\] This implies \(b_1=b_2=b_3=b_4=b_5=0\).

    1. Let \(W\) be the subspace of \(\mathbb{R}^5\) defined by \[W = \{(x_1, x_2, x_3, x_4, x_5) \in \mathbb{R}^5 \mid 6x_1 = x_2 \text{ and } x_3 + 2x_4 + 3x_5 = 0\}.\] Find a basis of \(W\).

    2. Extend the basis in (a) to a basis of \(\mathbb{R}^5\).

    3. Find a subspace \(V\) of \(\mathbb{R}^5\) such that \(\mathbb{R}^5 = W \oplus V\).

  2. Suppose \(V\) is finite-dimensional and \(U, W\) are subspaces of \(V\) such that \(V = U + W\). Prove that there exists a basis of \(V\) consisting of vectors in \(U \cup W\).

  3. Prove or give a counterexample: If \(p_0, p_1, p_2, p_3\) is a list in \(P_3(\mathbb{R})\) such that none of the polynomials \(p_0, p_1, p_2, p_3\) has degree 2, then \(p_0, p_1, p_2, p_3\) is not a basis of \(P_3(\mathbb{R})\).

  4. Suppose \(v_1, v_2, v_3, v_4\) is a basis of \(V\). Prove that \(v_1 + v_2, v_2 + v_3, v_3 + v_4, v_4\) is also a basis of \(V\).

  5. Prove or give a counterexample: If \(v_1, v_2, v_3, v_4\) is a basis of \(V\) and \(W\) is a subspace of \(V\) such that \(v_1, v_2 \in W\) and \(v_3 \notin W\) and \(v_4 \notin W\), then \(v_1, v_2\) is a basis of \(W\).

  6. Suppose \(v_1, \ldots, v_n\) is a list of vectors in \(V\). For \(i \in \{1, \ldots, n\}\), let \[w_i = v_1 + \cdots + v_i.\] Show that \(v_1, \ldots, v_n\) is a basis of \(V\) if and only if \(w_1, \ldots, w_n\) is a basis of \(V\).

  7. Suppose \(U\) and \(W\) are subspaces of \(V\) such that \(V = U \oplus W\). Suppose also that \(u_1, \ldots, u_m\) is a basis of \(U\) and \(w_1, \ldots, w_n\) is a basis of \(W\). Prove that \(u_1, \ldots, u_m, w_1, \ldots, w_n\) is a basis of \(V\).

  8. Suppose \(V\) is a real vector space. Show that if \(v_1, \ldots, v_n\) is a basis of \(V\) (as a real vector space), then \(v_1, \ldots, v_n\) is also a basis of the complexification \(V_{\mathbb{C}}\) (as a complex vector space). See Exercise 8 in Section 1B for the definition of the complexification \(V_{\mathbb{C}}\).

3.3 Exercise 2C (dimesion)