?Let ???? be the set of all ordered triples of real numbers, and consider the following

Let ???? be the set of all ordered triples of real numbers, and consider the following addition and scalar multiplication operations on \(\mathrm{u}=\left(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathrm{u}_{3}\right) \text { and } \mathrm{v}=\left(\mathrm{v}_{1}, \mathrm{v}_{2}, \mathrm{v}_{3}\right)\): \(\mathrm{u}+\mathrm{v}=\left(\mathrm{u}_{1}+\mathrm{v}_{1}, \mathrm{u}_{2}+\mathrm{v}_{2}, \mathrm{u}_{3}+\mathrm{v}_{3}\right), \mathrm{ku}=\left(\mathrm{ku}_{1}, 0,0\right)\) a. Compute u + v and ku for \(u=(3,-2,4), v=(1,5,-2)\), and \(k=-1\). b. In words, explain why ???? is closed under addition and scalar multiplication. c. Since the addition operation on ???? is the standard addition operation on \(R^{3}\), certain vector space axioms hold for ???? because they are known to hold for \(R^{3}\). Which axioms in Definition 1 of Section 4.1 are they? d. Show that Axioms 7, 8, and 9 hold. e. Show that Axiom 10 fails for the given operations. Equation Transcription: u = (u1, u2, u3) and v = (v1, v2, v3) u + v = (u1 + v1, u2 + v2, u3 + v3), ku = (ku1, 0, 0) u = (3, ?2, 4), v = (1, 5, ?2) k = ?1 ????3 Text Transcription: u = (u1, u2, u3) and v = (v1, v2, v3) u + v = (u1 + v1, u2 + v2, u3 + v3), ku = (ku1, 0, 0) u = (3, ?2, 4), v = (1, 5, ?2) k = ?1 ????3

In each part, the solution space of the system is a subspace of \(R^{3}\) and so must be a line through the origin, a plane through the origin, all of \(R^{3}\), or the origin only. For each system, determine which is the case. If the subspace is a plane, find an equation for it, and if it is a line, find parametric equations. a. \(0 x+0 y+0 z=0\) B.\(\begin{array}{r} 2 x-3 y+z=0 \\ 6 x-9 y+3 z=0 \\ -4 x+6 y-2 z=0 \end{array} \) c. \(\begin{array}{r} x-2 y+7 z=0 \\ -4 x+8 y+5 z=0 \\ 2 x-4 y+3 z=0 \end{array} \) d. \(\begin{array}{r} x+4 y+8 z=0 \\ 2 x+5 y+6 z=0 \\ 3 x+y-4 z=0 \end{array} \) Equation Transcription: ????3 0x + 0y + 0z = 0 2x ? 3y + z = 0 6x ? 9y + 3z = 0 ?4x + 6y ? 2z = 0 x ? 2y + 7z = 0 ?4x + 8y + 5z = 0 2x ? 4y + 3z = 0 x + 4y + 8z = 0 2x + 5y + 6z = 0 3x + y ? 4z = 0 Text Transcription: R^3 0x + 0y + 0z = 0 2x ? 3y + z = 0 6x ? 9y + 3z = 0 ?4x + 6y ? 2z = 0 x ? 2y + 7z = 0 ?4x + 8y + 5z = 0 2x ? 4y + 3z = 0 x + 4y + 8z = 0 2x + 5y + 6z = 0 3x + y ? 4z = 0

For what values of s is the solution space of \(x_{1}+x_{2}+s x_{3}=0\) \(x_{1}+s x_{2}+x_{3}=0\) \(\mathrm{sx}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=0\) the origin only, a line through the origin, a plane through the origin, or all of \(R^{3}\)? Equation Transcription: x1 + x2 + sx3 = 0 x1 + sx2 + x3 = 0 sx1 + x2 + x3 = 0 ????3 Text Transcription: x_1 + x_2 + sx_3 = 0 x_1 + sx_2 + x_3 = 0 sx_1 + x_2 + x_3 = 0 R^3

a. Express \((4a, a ? b, a + 2b)\) as a linear combination of \((4, 1, 1)\) and \((0, ?1, 2)\). b. Express \((3a + b + 3c, ?a + 4b ? c, 2a + b + 2c)\) as a linear combination of \((3, ?1, 2)\) and \((1, 4, 1)\). c. Express \((2a ? b + 4c, 3a ? c, 4b + c)\) as a linear combination of three nonzero vectors. Equation Transcription: (4a, a ? b, a + 2b) (4, 1, 1) (0, ?1, 2) (3a + b + 3c, ?a + 4b ? c, 2a + b + 2c) (3, ?1, 2) (1, 4, 1) (2a ? b + 4c, 3a ? c, 4b + c) Text Transcription: (4a, a ? b, a + 2b) (4, 1, 1) (0, ?1, 2) (3a + b + 3c, ?a + 4b ? c, 2a + b + 2c) (3, ?1, 2) (1, 4, 1) (2a ? b + 4c, 3a ? c, 4b + c)

Let ???? be the space spanned by \(f=\sin x\) and \(g=\cos x\). a. Show that for any value of \(\theta\), \(f_{1}=\sin (x+\theta)\) and \(g_{1}=\cos (x+\theta)\) are vectors in ????. b. Show that \(f_{1}\) and \(g_{1}\) form a basis for ????. Equation Transcription: Text Transcription: f = sin x g = cos x theta f_1= sin (x + theta ) g_1= cos (x + theta) f_1 g_1

a. Express \(v = (1, 1)\) as a linear combination of \(\mathrm{v}_{1}=(1,-1), \mathrm{v}_{2}=(3,0)\), and \(\mathrm{v}_{3}=(2,1)\) in two different ways. b. Explain why this does not violate Theorem 4.5.1. Equation Transcription: v = (1, 1) v1 = (1, ?1), v2 = (3, 0) v3 = (2, 1) Text Transcription: v = (1, 1) v_1 = (1, ?1), v_2 = (3, 0) v_3 = (2, 1)

Let ???? be an \(n \times n\) matrix, and let \(\mathrm{v}_{1}, \mathrm{v}_{2}, \ldots, \mathrm{v}_{\mathrm{n}}\) be linearly independent vectors in \(R^{n}\) expressed as \(n \times 1\) matrices. What must be true about ???? for \(A \mathrm{v}_{1}, A \mathrm{v}_{2}, \ldots, A \mathrm{v}_{\mathrm{n}}\) to be linearly independent? Equation Transcription: v1, v2, . . . , vn ????n ????v1, ????v2, . . . , ????vn Text Transcription: n x n v_1, v_2, . . . , v_n R^n n x 1 ????v_1, ????v_2, . . . , ????v_n

Must a basis for \(P_{\mathrm{n}}\) contain a polynomial of degree k for each \(\mathrm{k}=0,1,2, \ldots, \mathrm{n}\)? Justify your answer. Equation Transcription: ????n k = 0, 1, 2, . . . , n Text Transcription: P_n k = 0, 1, 2, . . . , n

For the purpose of this exercise, let us define a “checkerboard matrix” to be a square matrix \(A=\left[\mathrm{a}_{\mathrm{ij}}\right]\) such that \(\mathbf{a}_{i j}=\left\{\begin{array}{l} 1 \text { if } i+j \text { is even } \\ 0 \text { if } i+j \text { is odd } \end{array}\right. \) Find the rank and nullity of the following checkerboard matrices. a. The \(3 \times 3\) checkerboard matrix. b. The \(4 \times 4\) checkerboard matrix. c. The \(n \times n\) checkerboard matrix. Equation Transcription: ???? = [aij] aij = { Text Transcription: ???? = [a_ij] aij = {0 if i + j is odd1 if i + j is even 3 x 3 4 x 4 n x n

For the purpose of this exercise, let us define an “????-matrix” to be a square matrix with an odd number of rows and columns that has 0’s everywhere except on the two diagonals where it has 1’s. Find the rank and nullity of the following ????-matrices. (a) \(\left|\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right| \) (b) \(\left|\begin{array}{lllll} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{array}\right| \) (c) the ????-matrix of size \((2 n+1) \times(2 n+1)\) Equation Transcription: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>a</mi></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced><mi>b</mi></mfenced><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mfenced open="|" close="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> Text Transcription: |1 0 1 0 1 0 1 0 1| |1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1| (2n + 1) x (2n + 1)

In each part, show that the stated set of polynomials is a subspace of \(P_{\mathrm{n}}\) and find a basis for it. a. All polynomials in \(P_{\mathrm{n}}\) such that \(p(-x)=p(x)\). b. All polynomials in \(P_{\mathrm{n}}\) such that \(p(0)=p(1)\). Equation Transcription: ????n p(?x) = p(x) p(0) = p(1) Text Transcription: P_n p(?x) = p(x) p(0) = p(1)

(Calculus required) Show that the set of all polynomials in \(P_{\mathrm{n}}\) that have a horizontal tangent at \(x = 0\) is a subspace of \(P_{\mathrm{n}}\). Find a basis for this subspace. Equation Transcription: ????n x = 0 Text Transcription: P_n x=0

a. Find a basis for the vector space of all \(3 \times 3\) symmetric matrices. b. Find a basis for the vector space of all \(3 \times 3\) skew-symmetric matrices. Equation Transcription: Text Transcription: 3 × 3

Various advanced texts in linear algebra prove the following determinant criterion for rank: The rank of a matrix ???? is r if and only if ???? has some r × r submatrix with a nonzero determinant, and all square submatrices of larger size have determinant zero. [Note: A submatrix of ???? is any matrix obtained by deleting rows or columns of ????. The matrix ???? itself is also considered to be a submatrix of ????.] In each part, use this criterion to find the rank of the matrix. a. \(\left|\begin{array}{lll} 1 & 2 & 0 \\ 2 & 4 & -1 \end{array}\right| \) b. \(\left|\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \end{array}\right| \) c. \left|\begin{array}{lll} 1 & 0 & 1 \\ 2 & -1 & 3 \\ 3 & -1 & 4 \end{array}\right| d. \(\left|\begin{array}{lrll} 1 & -1 & 2 & 0 \\ 3 & 1 & 0 & 0 \\ -1 & 2 & 4 & 0 \end{array}\right| \) Equation Transcription: || || || || Text Transcription: r x r |1 2 0 2 4 -1| |1 2 3 2 4 6| |1 0 1 2 -1 3 3 -1 4| |1 -1 2 0 3 1 0 0 -1 2 4 0|

Use the result in Exercise 14 to find the possible ranks for matrices of the form \(\left|\begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & a_{16} \\ 0 & 0 & 0 & 0 & 0 & a_{26} \\ 0 & 0 & 0 & 0 & 0 & a_{36} \\ 0 & 0 & 0 & 0 & 0 & a_{46} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} \end{array}\right| \) Equation Transcription: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>a</mi><mn>16</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>a</mi><mn>26</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>a</mi><mn>36</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>a</mi><mn>46</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>51</mn></msub></mtd><mtd><msub><mi>a</mi><mn>52</mn></msub></mtd><mtd><msub><mi>a</mi><mn>53</mn></msub></mtd><mtd><msub><mi>a</mi><mn>54</mn></msub></mtd><mtd><msub><mi>a</mi><mn>55</mn></msub></mtd><mtd><msub><mi>a</mi><mn>56</mn></msub></mtd></mtr></mtable></mfenced></math> Text Transcription: |0 0 0 0 0 a_16 0 0 0 0 0 a_26 0 0 0 0 0 a_36 0 0 0 0 0 a_46 a_51 a_52 a_53 a_54 a_55 a_56|

Prove: If S is a basis for a vector space ????, then for any vectors u and v in ???? and any scalar k, the following relationships hold. a. \((u+v)_{S}=(u)_{S}+(v)_{S}\) b. \((\mathrm{ku})_{\mathrm{s}}=\mathrm{k}(\mathrm{u})_{\mathrm{s}}\) Equation Transcription: (u + v)s = (u)s + (v)s (ku)s = k(u)s Text Transcription: (u + v)???? = (u)???? + (v)???? (ku)???? = k(u)????

Let \(D_{\mathrm{k}}\), \(R_{\theta}\), and \(S_{\mathrm{k}}\) be a dilation of \(R^{2}\) with factor k, a counterclockwise rotation about the origin of \(R^{2}\) through an angle \(\theta\), and a shear of \(R^{2}\) by a factor k, respectively. a. Do \(D_{\mathrm{k}}\) and \(R_{\theta}\) commute? b. Do \(R_{\theta}\) and \(S_{\mathrm{k}}\) commute? c. Do \(D_{\mathrm{k}}\) and \(S_{\mathrm{k}}\) commute? Equation Transcription: ????k ????k ????2. Text Transcription: D_k R_theta S_k R^2. theta

A vector space ???? is said to be the direct sum of its subspaces ???? and ????, written \(V=U \oplus W\), if every vector in ???? can be expressed in exactly one way as \(v = u + w\), where u is a vector in ???? and w is a vector in ????. a. Prove that \(V=U \oplus W\) if and only if every vector in ???? is the sum of some vector in ???? and some vector in ???? and \(U \cap W=\{0\}\). b. Let ???? be the xy-plane and ???? the z-axis in \(R^{3}\). Is it true that \(R^{3}=U \oplus W\)? Explain. c. Let ???? be the xy-plane and ???? the yz-plane in \(R^{3}\). Can every vector in \(R^{3}\) be expressed as the sum of a vector in ???? and a vector in ????? Is it true that \(R^{3}=U \oplus W\)? Explain. Equation Transcription: ???? = ???????? v = u + w ???? ???? = {0} ????3 ????3 = ???????? Text Transcription: ???? = ???? oplos ???? v = u + w ???? cap ???? = {0} R^3 R^3 = ???? oplos ????