Chapter 28

Prove each of the following: If \(U\) is a subspace of \(V\), then \(\operatorname{dim} U \leq \operatorname{dim} V\).

Prove that \(\\{(0,0,0,1),(0,0,1,1),(0,1,1,1),(1,1,1,1)\\}\) is a basis of \(\mathbb{R}^{4}\).

Prove that \(\\{(a, b, c): 2 a-3 b+c=0\\}\) is a subspace of \(\mathbb{R}^{3}\).

Prove each of the following: If \(U\) is a subspace of \(V\), and \(\operatorname{dim} U=\operatorname{dim} V\), then \(U=V\).

If \(\mathrm{a}=(1,2,3,4)\) and \(\mathbf{b}=(4,3,2,1)\), explain why \(\\{\mathrm{a}, \mathbf{b}\\}\) may be extended to a basis of \(\mathbb{R}^{4}\). Then find a basis of \(\mathbb{R}^{4}\) which includes a and \(\mathbf{b}\).

Let \(T\) be a \(k\)-dimensional subspace of an \(n\)-dimensional space \(V\). Prove that an \((n-k)\)-dimensional subspace \(U\) exists such that \(V=T \oplus U\).

Prove: Suppose \(\operatorname{dim} U=\operatorname{dim} V ; h\) is an isomorphism (that is, a bijective homomorphism) iff \(h\) is injective iff \(h\) is surjective.

Prove each of the following: Any set of vectors containing 0 is linearly dependent.

Let \(A\) be the set of eight vectors \((x, y, z)\) where \(x, y, z=1,2\). Prove that \(A\) spans \(\mathbb{R}^{3}\), and find a subset of \(A\) which is a basis of \(\mathbb{R}^{3}\).

Prove that \(\\{f: f(1)=0\\}\) is a subspace of \(\mathscr{F}(\mathbb{R})\).

If \(T\) and \(U\) are arbitrary subspaces of \(V\), prove that $$ \operatorname{dim}(T+U)=\operatorname{dim} T+\operatorname{dim} U-\operatorname{dim}(T \cap U) $$

Prove: Any \(n\)-dimensional vector space \(V\) over \(F\) is isomorphic to the space \(F^{n}\) of all \(n\)-tuples of elements of \(F\)

Prove each of the following: The set \(\\{a\\}\), containing only one nonzero vector a, is linearly independent.

If \(\mathscr{P} \ell_{n}\) is the subspace of \(\mathscr{P} \ell\) consisting of all polynomials of degree \(\leq n\), prove that \(\left\\{1, x, x^{2}, \ldots, x^{n}\right\\}\) is a basis of \(\mathscr{P} \ell_{n}\). Then find another basis of \(\mathscr{P} \ell_{n}\).

Prove that \(\\{f: f\) is a constant on the interval \([0,1]\\}\) is a subspace of \(\mathscr{F}(\mathbb{R})\).

Prove that \(\|_{2}(R)\), the set of all \(2 \times 2\) matrices of real numbers, with matrix addition and the scalar multiplication $$ k\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)=\left(\begin{array}{ll} k a & k b \\ k c & k d \end{array}\right) $$ is a vector space over \(\mathbb{R}\).

Prove each of the following: Any subset of an independent set is independent. Any set of vectors containing a dependent set is dependent.

Find a basis for each of the following subspaces of \(\mathbb{R}^{3}\) : (a) \(S_{1}=\\{(x, y, z): 3 x-2 y+z=0\\}\) (b) \(S_{2}=\\{(x, y, z): x+y-z=0\) and \(2 x-y+z=0\\}\)

Prove that the set of all even functions [that is, functions \(f\) such that \(f(x)=f(-x)]\) is a subspace of \(\mathscr{F}(R)\). Is the same true for the set of all the odd functions [that is, functions \(f\) such that \(f(-x)=-f(x)]\) ?

Prove each of the following: If \(\\{\mathbf{a}, \mathbf{b}, \mathbf{c}\\}\) is linearly independent, so is \(\\{\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}, \mathbf{a}+\mathbf{c}\\}\)

Prove that the set of all polynomials of degree \(\leq n\) is a subspace of \(\mathscr{P} \ell\).

Prove each of the following: If \(\left\\{\mathbf{a}_{1}, \ldots, \mathbf{a}_{n}\right\\}\) is a basis of \(V\), so is \(\left\\{k_{1} \mathrm{a}_{1}, \ldots, k_{n} \mathrm{a}_{n}\right\\}\) for any nonzero scalars \(k_{1}, \ldots, k_{n}\)

Let \(U\) be the subspace of \(\mathscr{F}(\mathbb{R})\) spanned by \(\left\\{\cos ^{2} x, \sin ^{2} x, \cos 2 x\right\\} .\) Find the dimension of \(U\), and then find a basis of \(U\).

Find a basis for the subspace of \(\mathscr{P} \ell\) spanned by $$ \left\\{x^{3}+x^{2}+x+1, x^{2}+1, x^{3}-x^{2}+x-1, x^{2}-1\right\\} $$