Decide whether each of the following is a subspace ofM22. If so, provide a basis. If

Use the definition of a vector space V to prove the following: a. \(0 \mathbf{u}=\mathbf{0}\) for every \(\mathbf{u} \in V\). b. \(-\mathbf{u}=(-1) \mathbf{u}\) for every \(\mathbf{u} \in V\). (Hint: The distributive property 7 is all important.)

Decide whether the following sets of vectors are linearly independent. a. ,_ 1 0 0 1 _ , _ 0 1 1 0 _ , _ 1 1 1 1 _- M22 b. {f1, f2, f3} P1, where f1(t) = t , f2(t) = t + 1, f3(t) = t + 2 c. {f1, f2, f3} C (R), where f1(t) = 1, f2(t) = cos t , f3(t) = sin t d. {f1, f2, f3} C0(R), where f1(t) = 1, f2(t) = sin2 t , f3(t) = cos2 t e. {f1, f2, f3} C (R), where f1(t) = 1, f2(t) = cos t , f3(t) = cos 2t f. {f1, f2, f3} C (R), where f1(t) = 1, f2(t) = cos 2t , f3(t) = cos2 t

Decide whether each of the following is a subspace ofM22. If so, provide a basis. If not, give a reason. a. , A M22 : _ 1 2 _ N(A) - b. , A M22 : _ 1 2 _ C(A) - c. {A M22 : rank(A) = 1} d. {A M22 : rank(A) 1} e. {A M22 : A is in echelon form} f. , A M22 : A _ 1 2 1 1 _ = _ 1 2 1 1 _ A - g. , A M22 : A _ 1 2 1 1 _ = _ 1 2 1 1 _ AT -

What is the dimension of the vector space \(\mathcal{M}_{m \times n}\) ? Give a basis.

Check that the subsets defined in Example 2 are in fact subspaces of \(\mathcal{M}_{n \times n}\). Find their dimensions.

Decide whether each of the following is a subspace of C0(R). If so, provide a basis and determine its dimension. If not, give a reason. a. {f : f (1) = 2} b. {f P2 : + 1 0 f (t)dt = 0} c. {f C1(R) : f _ (t) + 2f (t) = 0 for all t} d. {f P4 : f (t) tf _ (t) = 0 for all t} e. {f P4 : f (t) tf _ (t) = 1 for all t} f. {f C2(R) : f __ (t) + f (t) = 0 for all t} g. {f C2(R) : f __ (t) f _ (t) 6f (t) = 0 for all t} h. {f C1(R) : f (t) = + t 0 f (s)ds for all t}

Suppose \(a_0(t), \ldots, a_{n-1}(t)\) are continuous functions. Prove that the set of solutions y(t) of the \(n^{\text {th }}\)-order differential equation \(y^{(n)}(t)+a_{n-1}(t) y^{(n-1)}(t)+\cdots+a_2(t) y^{\prime \prime}(t)+a_1(t) y^{\prime}(t)+a_0(t) y(t)=0\) is a subspace of \(\mathcal{C}^n(\mathbb{R})\). Here \(y^{(k)}(t)\) denotes the \(k^{\text {th }}\) derivative of the function y(t). (See Theorem 3.4 of Chapter 7 for an algorithm for finding those solutions when the functions \(a_j(t)\) are constants.)

Let \(\mathcal{M}_{n \times n}\) denote the vector space of all \(n \times n\) matrices. a. Let \(\mathcal{S} \subset \mathcal{M}_{n \times n}\) denote the set of symmetric matrices (those satisfying \(A^{\top}=A\) ). Show that \(\mathcal{S}\) is a subspace of \(\mathcal{M}_{a \times n}\). What is its dimension? b. Let \(\mathcal{K} \subset \mathcal{M}_{n \times n}\) denote the set of skew-symmetric matrices (those satisfying \(A^{\top}=-A)\). Show that \(\mathcal{K}\) is a subspace of \(\mathcal{M}_{n \times n}\). What is its dimension? c. Show that \(\mathcal{K}+\mathcal{K}=\mathcal{M}_{n \times n}\). (See Exercise 2.5.22.)

Define the trace of an n n matrixA(denoted trA) to be the sum of its diagonal entries: trA = )n i=1 aii . a. Show that trA = tr(AT). b. Show that tr(A + B) = trA + trB and tr(cA) = c trA for any scalar c. c. Prove that tr(AB) = tr(BA). (Hint: 'n k=1 'n _=1 ck_ = 'n _=1 'n k=1 ck_.) d. If A,B Mnn, define _A,B_ = tr(ATB). Check that this is an inner product on Mnn. e. Check that if A is symmetric and B is skew-symmetric, then _A,B_ = 0. (Hint: Use the properties to show that _A,B_ = _B,A_.) f. Deduce that the subspaces of symmetric and skew-symmetric matrices (see Exercise 8) are orthogonal complements inMnn.

(See Exercise 9 for the relevant definitions.) Define \(V=\left\{A \in \mathcal{M}_{n \times n}: \operatorname{tr} A=0\right\}\). a. Show that V is a subspace of \(\mathcal{M}_{n \times n}\). b. Find a basis for \(V^{\perp}\) (using the inner product defined in Exercise 9).

Here is a sketch of the algebra result mentioned in the text. Let p be a polynomial of degree k, that is, p(t) = akt k + ak1t k1 + +a1t + a0, where a0, . . . , ak R and ak _= 0. a. Prove the root-factor theorem: c is a root of p, i.e., p(c) = 0, if and only if p(t) = (t c)q(t) for some polynomial q of degree k 1. (Hint: When you divide p(t) by t c, the remainder should be p(c). Why?) b. Show that p has at most k roots.

a. Suppose b, c, and d are distinct real numbers. Show that the matrix \(\left[\begin{array}{ccc}1 & 1 & 1 \\b & c & d \\b^2 & c^2 & d^2\end{array}\right]\) is nonsingular. b. Suppose a, b, c, and d are distinct real numbers. Show that the matrix \(\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\a & b & c & d \\a^2 & b^2 & c^2 & d^2 \\a^3 & b^3 & c^3& d^3\end{array}\right]\) is nonsingular. (Hint: Subtract a times the third row from the fourth, a times the second row from the third, and a times the first row from the second.) c. Suppose \(t_1, \ldots, t_{k+1}\) are distinct. Prove that the matrix \(\left[\begin{array}{cccc}1 & 1 & \ldots & 1 \\t_1 & t_2 & \ldots & t_{k+1} \\t_1^2 & t_2^2 & \ldots & t_{k+1}^2 \\\vdots & \vdots & \ddots & \vdots \\t_1^k & t_2^k & \ldots & t_{k+1}^k\end{array}\right]\) is nonsingular. (Hint: Iterate the trick from part b. If you know mathematical induction, this would be a good place to try it.)

a. Prove that \(\operatorname{dim} \mathcal{M}_{n \times n}=n^2\). b. Let \(A \in \mathcal{M}_{n \times n}\). Show that there are scalars \(c_0, c_1, \ldots, c_{n^2}\), not all 0 , so that \(c_0 I_n+\) \(c_1 A+c_2 A^2+\cdots+c_{n^2} A^{n^2}=O\). (That is, there is a nonzero polynomial p of degree at most \(n^2\) so that p(A) =O.)

Let g(t) = 1. Using the inner product defined in Example 10(c), find the orthogonal complement of \(\operatorname{Span}(\mathrm{g})\) in a. \(\mathcal{P}_1 \subset \mathcal C^0([-1,1])\) b. \(\mathcal{P}_1 \subset \mathcal{C}^0([0,1])\) c. \(\mathcal{P}_2 \subset \mathcal C^0([-1,1])\) d. \(\mathcal{P}_2 \subset \mathcal C^0([0,1])\)

Let g1(t) = 1 and g2(t) = t . Using the inner product defined in Example 10(c), find the orthogonal complement of Span (g1, g2) in a. P2 C0([1, 1]) b. P2 C0([0, 1]) c. P3 C0([1, 1]) d. P3 C0([0, 1])

Let \(g_1(t)=t-1\) and \(g_2(t)=t^2+t\). Using the inner product on \(P_2\subset C^0\left ( \left [ 0,1 \right ] \right )\) defined in Example 10(c), find the orthogonal complement of Span \(\left ( g_1,g_2 \right )\) .

Let and . Using the inner product on defined in Example 10(b) with , and , find a basis for the orthogonal complement of Span .

Show that for any positive integer n, the functions 1, cos t , sin t , cos 2t , sin 2t, . . . , cos nt, sin nt are orthogonal in C ([, ]) C0([, ]) (using the inner product defined in Example 10(c)).

Using the inner product defined in Example 10(c), let \(V=\mathcal C^0([a, b])\), and let \(W= \left\{f \in V: \int_a^b f(t) d t=0\right\}\). a. Prove that W is a subspace of V. b. Prove that \(W^{\perp}\) is the subspace of constant functions. c. Prove or disprove: \(W+W^{\perp}=V\).

Prove that the following are subspaces of \(\mathbb{R}^\omega\). a. \(\left\{\mathbf{x}\right.\) : there is a constant C such that \(\left|x_k\right| \leq C\) for all \(\left.k\right\}\) b. \(\left\{\mathbf{x}: \lim _{k \rightarrow \infty} x_k\right.\) exists \(\}\) c. \(\left\{\mathbf{x}: \lim _{k \rightarrow \infty} x_k=0\right\}\) d. \(\left\{\mathbf{x}: \sum_{k=1}^{\infty} x_k\right.\) exists \(\}\) e. \(\left\{\mathbf{x}: \sum_{k=1}^{\infty}\left|x_k\right|\right.\) exists \(\}\) (Hint: Remember that \(|a+b| \leq|a|+|b|\) for all \(a, b \in \mathbb{R}\).)

The subspace _2 R defined by _2 = {x R : ' k=1 x2 k exists} is an inner product space with inner product defined by _x, y_ = ' k=1 xkyk. (That this sum makes sense follows by taking the limit of the Cauchy-Schwarz Inequality.) Let V = {x _2 : there is an integer n such that xk = 0 for all k > n}. Show that V is a (proper) subspace of _2 and that V = {0}. It follows that (V ) = _2, so Proposition 3.6 need not hold in infinite-dimensional spaces.