5.E: Exercises for Chapter 5

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Calculational Exercises

1. Show that the vectors \(v_1 = (1, 1, 1), v_2 = (1, 2, 3)\), and \(v_3 = (2, −1, 1)\) are linearly independent in \(\mathbb{R}^3\). Write \(v = (1, −2, 5)\) as a linear combination of \(v_1 , v_2\), and \(v_3\).

2. Consider the complex vector space \(V = \mathbb{C}^3\) and the list \((v_1 , v_2 , v_3 )\) of vectors in \(V\) , where

\[v_1 = (i, 0, 0),~ v_2 = (i, 1, 0),~ v_3 = (i, i, −1).\]

(a) Prove that \(span(v_1 , v-2 , v_3 ) = V.\)
(b) Prove or disprove: \((v_1 , v_2 , v_3)\) is a basis for \(V.\)

3. Determine the dimension of each of the following subspaces of \(\mathbb{F}^4\) .
(a) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = 0\}.\)
(b) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 \}.\)
(c) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 , x_3 = x_1 − x_2 \}.\)
(d) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 , x_3 = x_1 − x_2 , x_3 + x_4 = 2x_1 \}.\)
(e) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_1 = x_2 = x_3 = x_4 \}.\)

4. Determine the value of \(\lambda \in \mathbb{R}\) for which each list of vectors is linear dependent.
(a) \(((\lambda, −1, −1), (−1, \lambda, −1), (−1, −1, \lambda))\) as a subset of \(\mathbb{R}^3.\)
(b) \(sin2 (x), cos(2x), \lambda\) as a subset of \(\cal{C}(\mathbb{R}).\)

5. Consider the real vector space \(V = \mathbb{R}^4.\) For each of the following ﬁve statements, provide either a proof or a counterexample.
(a) \(dim V = 4.\)
(b) \(span((1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1)) = V.\)
(c) The list \(((1, −1, 0, 0), (0, 1, −1, 0), (0, 0, 1, −1), (−1, 0, 0, 1))\) is linearly independent.
(d) Every list of four vectors \(v_1 , \ldots, v_4 \in V\) , such that \(span(v_1 , \ldots, v_4 ) = V\) , is linearly independent.
(e) Let \(v_1\) and \(v_2\) be two linearly independent vectors in \(V\) . Then, there exist vectors \(u, w \in V\) , such that \((v_1 , v_2 , u, w)\) is a basis for \(V.\)

Proof-Writing Exercises

1. Let \(V\) be a vector space over \(\mathbb{F}\) and define \(U = span(u_1, u_2, \ldots ,u_n)\), where for each
\(i = 1, \ldots ,n, u_i \in V \). Now suppose \(v \in U\). Prove

\[U = span(v, u_1, u_2, \ldots ,u_n) .\]

2. Let \(V\) be a vector space over \(\mathbb{F}\), and suppose that the list \((v_1 , v_2 , . . . , v_n )\) of vectors spans \(V\) , where each \(v_i \in V\) . Prove that the list

\[(v_1 − v_2 , v_2 − v_3 , v_3 − v_4 , \ldots , v_{n−2} − v_{n−1} , v_{n−1} − v_n , v_n )\]

also spans \(V.\)

3. Let \(V\) be a vector space over \(\mathbb{F}\), and suppose that \((v_1 , v_2 , \ldots, v_n)\) is a linearly independent list of vectors in \(V\) . Given any \(w \in V\) such that

\[(v_1 + w, v_2 + w, \ldots , v_n + w)\]

is a linearly dependent list of vectors in \(V\) , prove that \(w \in span(v_1 , v_2 , \ldots, v_n).\)

4. Let \(V\) be a ﬁnite-dimensional vector space over \(\mathbb{F}\) with \(dim(V ) = n\) for some \(n \in \mathbb{Z}_+\). Prove that there are \(n\) one-dimensional subspaces \(U_1 , U_2 , \ldots , U_n\) of \(V\) such that

\[V = U_1 \oplus U_2 \oplus \cdots \oplus U_n .\]

5. Let \(V\) be a ﬁnite-dimensional vector space over \(\mathbb{F}\), and suppose that \(U\) is a subspace of \(V\) for which \(dim(U) = dim(V ).\) Prove that \(U = V.\)

6. Let \(\mathbb{F}_m [z] \) denote the vector space of all polynomials with degree less than or equal to \(m \in \mathbb{Z}_+\) and having coeﬃcient over \(\mathbb{F}\), and suppose that\( p_0 , p_1 , \ldots , p_m \in \mathbb{F}_m [z]\) satisfy \(p_j (2) = 0\). Prove that \((p_0 , p_1 , \ldots , p_m )\) is a linearly dependent list of vectors in \(\mathbb{F}_m [z].\)

7. Let \(U\) and \(V\) be ﬁve-dimensional subspaces of \(\mathbb{R}^9\) . Prove that \(U \cap V = \{0\}.\)

8. Let \(V\) be a ﬁnite-dimensional vector space over \(\mathbb{F},\) and suppose that \(U_1 , U_2 , \ldots, U_m\) are any \(m\) subspaces of \(V\) . Prove that

\[dim(U_1 + U_2 + \cdots + U_m ) \leq dim(U_1 ) + dim(U_2 ) + \cdots + dim(U_m ).\]

Contributors

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