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Section 5.3 Vector Spaces: Homework
Exercises Exercises
Exercise Group. Prove the following sets are vector spaces over \(\R\) under the given operations.
1. The set of all convergent real sequences under term-wise addition and scalar multiplication.
2. The set of matrices, \(\R^{m\times n}\text{,}\) under matrix addition and scalar multiplication.
Exercise Group. Prove that the following sets, \(S\text{,}\) are subspaces of the given vector space, \(V\text{.}\)
3.
\(S=V\text{.}\) (I.e., prove that \(V\) is a subspace of itself)
4.
\(S=\{\vec{0}\}\text{,}\) \(V\text{.}\) (I.e., prove that \(\{\vec{0}\}\) is a subspace of any vector space.)
5. For any given \(n\in\N,\,S=\mathcal{L}=\left\{L\in\R^{n\times n}\left|\right.
L_{ij}=0\,\text{ if } i\lt j\right\}\) of lower-triangular matrices, \(V=\R^{n\times n}\text{.}\)
6. For any given \(n\in\N,\,S=\mathcal{U}:=\left\{U\in\R^{n\times n}\left|\right.
U_{ij}=0\,\text{ if } i>j\right\}\) of upper-triangular matrices, \(V=\R^{n\times n}\text{.}\)
7.
\(S=\left\{\left.\left(\begin{array}{c}x\\y\\z \end{array} \right)\right|x+y+z=0\right\}\text{,}\) \(V=\R^3\text{.}\)
8.
\(S=\left\{\left.\left(\begin{array}{c}x\\2x\\z \end{array} \right)\right|x,z\in\R\right\}\text{,}\) \(V=\R^3\text{.}\)
9.
\(S=\left\{\left.\left(\begin{array}{c}x\\y\\2x+3y \end{array} \right)\right|x,y\in\R\right\}\text{,}\) \(V=\R^3\text{.}\)
10. Let \(A\in\R^{n\times n}\text{.}\) \(S=N(A)=\left\{\left.\vec{x}\right|A\vec{x}=\vec{0}\right\}\text{,}\) \(V=\R^n\text{.}\)
11.
\(S=\left\{\left.\left(\begin{array}{cc}0\amp a\\b\amp 0 \end{array} \right)\right|a,b\in\R\right\}\text{,}\) \(V=\R^{2\times 2}\text{.}\)
Exercise Group. Prove that the following sets, \(S\text{,}\) are not subspaces of the given vector space, \(V\text{.}\)
12.
\(S=\left\{\left.\left(\begin{array}{c}x^2\\x \end{array} \right)\right|x\in\R\right\}\text{,}\) \(V=\R^2\text{.}\)
13.
\(S=\left\{\left.\left(\begin{array}{c}x^3\\x^4 \end{array} \right)\right|x\in\R\right\}\text{,}\) \(V=\R^2\text{.}\)
14.
\(S=\left\{\left.\left(\begin{array}{c}x\\y\\z \end{array} \right)\right|xyz=0\right\}\text{,}\) \(V=\R^3\text{.}\)
15.
\(S=\left\{\left.\left(\begin{array}{c}x\\y\\z \end{array} \right)\right|x,y,z\in\R,\,y\ge z\right\}\text{,}\) \(V=\R^3\text{.}\)
16.
\(S=\left\{\left.\left(\begin{array}{c}x\\y\\z \end{array} \right)\right|x\ne0\in\R\right\}\text{,}\) \(V=\R^3\text{.}\)
17.
\(S=\left\{A\in\R^{3\times 3}|\det(A)>1\right\}\text{,}\) \(V=\R^{3\times 3}\text{.}\)
18.
\(S=SL_2(\R):=\left\{A\in\R^{2\times 2}\,|\,ad-bc=1\right\}\text{,}\) \(V=\R^{2\times 2}\text{.}\)
19. Theorem 5.1.29 .20. Theorem 5.1.30 .21. Theorem 5.1.32 .22. Theorem 5.1.33 .23. 24. Theorem 5.1.35 .25. Theorem 5.1.36 .26. Theorem 5.1.37 .
Exercise Group.
27. Prove that the set of second-degree polynomials defined on the closed interval \([-1,1]\) is a vector space.
28. Let \(V\) be the set of second-degree polynomials defined on \([-1,1].\) Prove that \(\ip{p(x)}{q(x)}:=\displaystyle{\int_{-1}^1p(x)q(x)\,dx}\) is indeed an inner product on \(V\) and state the induced norm on \(V.\)
