?In Exercises 11–12, show that the matrices are orthogonal with respect to the standard

Let the vector space \(\mathrm{P} 4\) have the evaluation inner product at the points \(-2,-1,0,1,2\) and let \(p=p(x)=x+x^{3}\) \(\text { and } q=q(x)=1+x^{2}+x^{4}\) a. Compute \(\langle\mathrm{p}, \mathrm{q}\rangle,\|\mathrm{p}\|, \text { and }\|\mathrm{q}\|\). b. Verify that the identities in Exercises 44 and 45 hold for the vectors \(p \text { and } q\). Equation Transcription: Text Transcription: P4 -2,-1,0,1,2 p=p(x)=x+x3 and q=q(x)=1+x2+x4 ?p,q ?,||p||, and ||q|| p and q

Let the vector space M33 have the standard inner product and let \(\mathbf{u}=U=\left[\begin{array}{rrr} 1 & -2 & 3 \\ -2 & 4 & 1 \\ 3 & 1 & 0 \end{array}\right] \text { and } \mathbf{v}=V=\left[\begin{array}{rrr} 2 & -1 & 0 \\ 1 & 4 & 3 \\ 1 & 0 & 2 \end{array}\right]\) a. Use Formula (8) to compute \(\langle\mathrm{u}, \mathrm{v}\rangle, \text { llull, and } \| \mathrm{vll}\). b. Verify that the identities in Exercises 44 and 45 hold for the vectors \(u \text { and } v\). Equation Transcription: [ ] and [ ] Text Transcription: u=U=[ 3 1 0-2 4 1 1 -2 3 ] and v=V=[ 1 0 2 1 4 3 2 -1 0 ] \langle u,v \rangle ,||u||, and ||v|| u and v

In Exercises 1–2, find the cosine of the angle between the vectors with respect to the Euclidean inner product. a. \(u=(1,-3), v=(2,4)\) b. \(\mathrm{u}=(-1,5,2), \mathrm{v}=(2,4,-9)\) c. \(u=(1,0,1,0), v=(-3,-3,-3,-3)\) Equation Transcription: Text Transcription: u=(1,-3), v =(2,4) u=(-1,5,2), v=(2,4,-9) u=(1,0,1,0), v=(-3,-3,-3,-3)

In Exercises 1–2, find the cosine of the angle between the vectors with respect to the Euclidean inner product. a. \(u=(-1,0), v=(3,8)\) b. \(u=(4,1,8), v=(1,0,-3)\) c. \(\mathrm{u}=(2,1,7,-1), \mathrm{v}=(4,0,0,0)\) Equation Transcription: Text Transcription: u=(-1,0), v=(3,8) u=(4,1,8), v=(1,0,-3) u=(2,1,7,-1), v=(4,0,0)

In Exercises 3–4, find the cosine of the angle between the vectors with respect to the standard inner product on P2. \(p=-1+5 x+2 x^{2}, q=2+4 x-9 x^{2}\) Equation Transcription: Text Transcription: p=-1+5 x+2 x^2, q=2+4 x-9 x^2

In Exercises 3–4, find the cosine of the angle between the vectors with respect to the standard inner product on P2. \(p=x-x^{2}, q=7+3 x+3 x^{2}\) Equation Transcription: Text Transcription: p=x-x^2,q=7+3x+3x^2

In Exercises 5–6, find the cosine of the angle between \(A \text { and } B\) with respect to the standard inner product on \(\mathrm{M}_{22}\) \(A=\left[\begin{array}{rr} 2 & 6 \\ 1 & -3 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & 2 \\ 1 & 0 \end{array}\right]\) Equation Transcription: [], [] Text Transcription: A and B M_22 A=[ 1 -3 2 6 ], B=[ 1 0 3 2 ]

In Exercises 5–6, find the cosine of the angle between \(A \text { and } B\) with respect to the standard inner product on \(\mathrm{M}_{22}\). \(A=\left[\begin{array}{rr} 2 & 4 \\ -1 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} -3 & 1 \\ 4 & 2 \end{array}\right]\) Equation Transcription: [], [] Text Transcription: A and B M_22 A=[ -1 3 2 4 ], B=[ 4 2 -3 1 ]

In Exercises 7–8, determine whether the vectors are orthogonal with respect to the Euclidean inner product. a. \(u=(-1,3,2), v=(4,2,-1)\) b. \(\mathrm{u}=(-2,-2,-2), \mathrm{v}=(1,1,1)\) c. \(\mathrm{u}=(\mathrm{a}, \mathrm{b}), \mathrm{v}=(-\mathrm{b}, \mathrm{a})\) Equation Transcription: Text Transcription: u=(-1,3,2), v=(4,2,-1) u=(-2,-2,-2), v =(1,1,1) u=(a,b), v=(-b,a)

In Exercises 7–8, determine whether the vectors are orthogonal with respect to the Euclidean inner product. a. \(\mathrm{u}=(\mathrm{u} 1, \mathrm{u} 2, \mathrm{u} 3), \mathrm{v}=(0,0,0)\) b. \(\mathrm{u}=(-4,6,-10,1), \mathrm{v}=(2,1,-2,9)\) c. \(\mathrm{u}=(\mathrm{a}, \mathrm{b}, \mathrm{c}), \mathrm{v}=(-\mathrm{c}, 0, \mathrm{a})\) Equation Transcription: Text Transcription: u=(u1,u2,u3), v=(0,0,0) u=(-4,6,-10,1), v=(2,1,-2,9) u=(a,b,c), v=(-c,0,a)

In Exercises 9–10, show that the vectors are orthogonal with respect to the standard inner product on \(P_{2}\). \(p=-1-x+2 x^{2}, q=2 x+x^{2}\) Equation Transcription: Text Transcription: P_2 p=-1-x+2x^2, q=2x+x^2

In Exercises 9–10, show that the vectors are orthogonal with respect to the standard inner product on \(P_{2}\). \(p=2-3 x+x^{2}, q=4+2 x-2 x^{2}\) Equation Transcription: Text Transcription: P_2 p=2-3x+x^2,q=4+2x-2x^2

In Exercises 11–12, show that the matrices are orthogonal with respect to the standard inner product on \(\mathrm{M}_{22}\). \(U=\left[\begin{array}{rr} 2 & 1 \\ -1 & 3 \end{array}\right], \quad V=\left[\begin{array}{rr} -3 & 0 \\ 0 & 2 \end{array}\right]\) Equation Transcription: [],[] Text Transcription: M_22 U=[ -1 3 2 1 ],V=[ 0 2 -3 0 ]

In Exercises 11–12, show that the matrices are orthogonal with respect to the standard inner product on \(\mathrm{M}_{22}\). \(U=\left[\begin{array}{ll} 5 & -1 \\ 2 & -2 \end{array}\right], V=\left[\begin{array}{rr} 1 & 3 \\ -1 & 0 \end{array}\right]\) Equation Transcription: [],[] Text Transcription: M_22 U=[ 2 -2 5 -1 ],V=[-1 0 1 3 ]

In Exercises 13–14, show that the vectors are not orthogonal with respect to the Euclidean inner product on \(\mathrm{R}^{2}\), and then find a value of \(\text { k }\) for which the vectors are orthogonal with respect to the weighted Euclidean inner product \(\langle u, v\rangle=2 u_{1}+v_{1}+k u_{2} v_{2}\). \(\mathrm{u}=(1,3), \mathrm{v}=(2,-1)\) Equation Transcription: Text Transcription: R2 k \langle u,v \rangle =2u1+v1+ku2v2 u=(1,3), v=(2,-1)

In Exercises 13–14, show that the vectors are not orthogonal with respect to the Euclidean inner product on \(\mathrm{R}^{2}\), and then find a value of \(\text { k }\) for which the vectors are orthogonal with respect to the weighted Euclidean inner product \(\langle u, v\rangle=2 u_{1}+v_{1}+k u_{2} v_{2}\). \(\mathrm{u}=(2,-4), \mathrm{v}=(0,3)\) Equation Transcription: Text Transcription: R2 k \langle u,v \rangle =2u1+v1+ku2v2 u=(2,-4), v=(0,3)

If the vectors \(\mathrm{u}=(1,2) \text { and } \mathrm{v}=(2,-4)\) are orthogonal with respect to the weighted Euclidean inner product \(\langle\mathrm{u}, \mathrm{v}\rangle=\mathrm{w}_{1} \mathrm{u}_{1} \mathrm{v}_{1}+\mathrm{w}_{2} \mathrm{u}_{2} \mathrm{v}_{2}\) what must be true of the weights \(\mathrm{w}_{1} \text { and } \mathrm{w}_{2}\)? Equation Transcription: Text Transcription: u=(1,2) and v=(2,-4) \langle u,v \rangle =w1u1v1+w2u2v2 w1 and w2

Let \(R^{4}\) have the Euclidean inner product. Find two unit vectors that are orthogonal to all three of the vectors \(u=(2,1,-4,0), v=(-1,-1,2,2), \text { and } w=(3,2,5,4)\). Equation Transcription: Text Transcription: R^4 u=(2,1,-4,0), v=(-1,-1,2,2), and w=(3,2,5,4)

Do there exist scalars k and l such that the vectors \(p_{1}=2+k x+6 x^{2}, p_{2}=l+5 x+3 x^{2}, p_{3}=1+2 x+3 x^{2}\) are mutually orthogonal with respect to the standard inner product on P2? Equation Transcription: Text Transcription: p1=2+kx+6x^2, p2=l+5x+3x^2,p3=1+2x+3x^2 P_2

Show that the vectors \(u=\left[\begin{array}{l} 3 \\ 3 \end{array}\right] \text { and } v=\left[\begin{array}{c} 5 \\ -8 \end{array}\right] \) are orthogonal with respect to the inner product on R2 that is generated by the matrix \(\mathrm{A}=\left[\begin{array}{ll} 2 & 1 \\ 1 & 1 \end{array}\right] \) [See Formulas (5) and (6) of Section 6.1.] Equation Transcription: u= [] and v=[] A= [] Text Transcription: u= [3 3] and v=[-8 5] A= [1 1 2 1]

Let \(\mathbf{P}_{2}\) have the evaluation inner product at the points \(x_{0}=-2, x_{1}=0, x_{2}=2\) Show that the vectors \(\mathbf{p}=\mathrm{x}\) and \(\mathbf{q}=\mathrm{x}^{2}\) are orthogonal with respect to this inner product. Equation Transcription: P2 x0 = ?2, x1 = 0, x2 = 2 p = x q = x2 Text Transcription: P_2 x_0 = ?2, x_1 = 0, x_2 = 2 p = x q = x^2

Let \(\mathrm{M}_{22}\) have the standard inner product. Determine whether the matrix A is in the subspace spanned by the matrices U and V. \(A=\left[\begin{array}{ll} -1 & 1 \\ 0 & 2 \end{array}\right], \quad U=\left[\begin{array}{rr} 1 & -1 \\ 3 & 0 \end{array}\right], \quad V=\left[\begin{array}{cc} 4 & 0 \\ 9 & 2 \end{array}\right] \) Equation Transcription: M22 A= [], U=[], V=[] Text Transcription: M_22 A= [0 2 -1 1], U=[3 0 1 -1], V=[9 2 4 0]

In Exercises 21-24, confirm that the Cauchy-Schwarz, inequality holds for the given vectors using the stated inner product. \(u=(1,0,3), v=(2,1,-1)\) using the weighted Euclidean inner product \(\langle\mathrm{u}, \mathrm{v}\rangle=2 u_{1} v_{1}+3 u_{2} v_{2}+u_{3} v_{3}\) in \(R^{3}\) Equation Transcription: ?u,v? Text Transcription: ?u,v? R^3

In Exercises 21-24, confirm that the Cauchy-Schwarz, inequality holds for the given vectors using the stated inner product. \(\mathrm{U}=\left[\begin{array}{cc} -1 & 2 \\ 6 & 1 \end{array}\right] \text { and } \mathrm{V}=\left[\begin{array}{ll} 1 & 0 \\ 3 & 3 \end{array}\right] \) using the standard inner product on \(M_{22}\). Equation Transcription: U=[] and V=[] Text Transcription: U=[-1 2 6 1] and V=[1 0 3 3] M_22

In Exercises 21-24, confirm that the Cauchy-Schwarz, inequality holds for the given vectors using the stated inner product. \(p=-1+2 x+x^{2}\) and \(q=2-4 x^{2}\) using the standard inner product on \(P_{2}\) Equation Transcription: Text Transcription: p=-1+2x+x^2 q=2-4x^2 P_2

In Exercises 21-24, confirm that the Cauchy-Schwarz, inequality holds for the given vectors using the stated inner product. The vectors \(u=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] \text { and } v=\left[\begin{array}{c} 1 \\ -1 \end{array}\right] \) with respect to the inner product in Exercise 18 . Equation Transcription: u=[] and v=[] Text Transcription: u=[1 1] v=[1 -1]

Let \(R^{4}\) have the Euclidean inner product, and suppose that \(u=(-1,1,0,2)\). Determine whether the vector u is orthogonal to the subspace spanned by the vectors \(w_{1}=(1,-1,3,0)\) and \(w_{2}=(4,0,9,2)\) a. Let W be the line in \(R^{2}\) with equation \(y=2 x\). Find an equation for \(W^{\perp}\). b. Let W be the plane in \(R^{3}\) with equation \(x-2 y-3 z=0\). Find parametric equations for \(W^{\perp}\). Equation Transcription: Text Transcription: R^4 u=(-1,1,0,2) w_1=(1,-1,3,0) w_2=(4,0,9,2) R^2 y=2x W^prep R^3 x-2y-3z=0

Let \(P_{3}\) have the standard inner product, and let \(p=-1-x+2 x^{2}+4 x^{3}\) Determine whether the polynomial p is orthogonal to the subspace spanned by the polynomials \(w_{1}=2-x^{2}+x^{3}\) and \(w_{2}=4 x-2 x^{2}+2 x^{3}\) Equation Transcription: Text Transcription: P_3 p=-1-x+2x^2+4x^3 w_1=2-x^2+x^3 w_2=4x-2x^2+2x^3

In Exercises , find a basis for the orthogonal complement of the subspace of \(R^{n}\) spanned by the vectors \(v_{1}=(1,4,5,2), v_{2}=(2,1,3,0), v_{3}=(-1,3,2,2)\) Equation Transcription: Text Transcription: R^n v_1=(1,4,5,2),v_2=(2,1,3,0),v_3=(-1,3,2,2)

Let \(R^{4}\) have the Euclidean inner product. a. Find a vector in \(R^{4}\) that is orthogonal to \(\mathbf{u}_{1}=(1,0,0,0)\) and \(\mathbf{u}_{4}=(0,0,0,1)\) and makes equal angles with the \(\mathbf{u}_{2}=(0,1,0,0)\) and \(\mathbf{u}_{3}=(0,0,1,0)\) b. Find a vector \(x=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\) of length \(1\) that is orthogonal to \(\mathbf{u}_{1}\) and \(\mathbf{u}_{4}\) above and such that the cosine of the angle between \(x\) and \(\mathbf{u}_{2}\) is twice the cosine of the angle between \(x\) and \(\(\mathbf{u}_{1}\)\). Equation Transcription: Text Transcription: R^4 u_1 = (1,0,0,0) u_4 = (0,0,0,1) u_2 = (0,1,0,0) u_3 = (0,0,1,0) x = (x_1,x_2,x_3,x_4) 1 u_1 u_4 x u_2 u_3

Prove: If \(\langle\mathbf{u}, v\rangle\) is the Euclidean inner product on \(R^{n}\) , and if \(A\) is an \(n \times n\) matrix, then \(\langle\mathbf{u}, A v\rangle=\left\langle A^{T} \mathbf{u}, v\right\rangle\) Hint Use the fact that \(\langle\mathbf{u}, v\rangle=\mathbf{u} \cdot v=v^{T} \mathbf{u}\)] Equation Transcription: Text Transcription: <u,v> R^n A N times n <u, Av> = <A^T u, v> <u,v> = u cdot v = v^T u

Let \(M_{22}\) have the inner product \(\langle U, V\rangle=\operatorname{tr}\left(U^{T} V\right)=t r\left(V^{T} U\right)\) that was defined in Example \(6\) of Section \(6.1\). Describe the orthogonal complement of a. the subspace of all diagonal matrices. b. the subspace of symmetric matrices. Equation Transcription: Text Transcription: M_22 <U,V> = tr(U^T V) = tr(V^T U) 6 6.1

Let \(A x=0\) be a system of \(m\) equations in \(n\) unknowns. Show that \(\mathbf{x}=\left[\begin{array}{c}x_{1} \\x_{2} \\\vdots \\x_{n}\end{array}\right]\) is a solution of this system if and only if \(x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is orthogonal to every row vector of \(A\) with respect to the Euclidean inner product on \(R^{n}\) Equation Transcription: Text Transcription: Ax=0 m n x=[x_1 \\x_2 \\\vdots \\x_n] x=(x_1, x_2,..., x_n) A R^n

Use the Cauchy-Schwarz inequality to show that if \(a_{1}, a_{2}, \ldots, a_{n}\) are positive real numbers, then \(\left(a_{1}+a_{2}+\cdots+a_{n}\right)\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right) \geq n^{2}\) Equation Transcription: Text Transcription: a_1, a_2, ..., a_n (a_1+a_2+ cdots +a_n)(1/a_1 + 1/a_2 + cdots +1/a_n) geq n^2

Show that if \(x\) and \(y\) are vectors in an inner product space and \(c\) is any scalar, then \(\|c x+y\|^{2}=c^{2}\|x\|^{2}+2 c\langle x, y\rangle+\|y\|^{2}\) Equation Transcription: Text Transcription: x y c |cx + y^2|^2 = c^2 |x|^2 + 2c <x,y> + |y|^2

Let \(R^{3}\) have the Euclidean inner product. Find two vectors of length \(1\) each of which is orthogonal to all three of the vectors \(\mathbf{u}_{1}=(1,1,-1), \mathbf{u}_{2}=(-2,-1,2)\) and \(\mathbf{u}_{3}=(-1,0,1)\). Equation Transcription: Text Transcription: R^3 1 u_1 = (1, 1, -1) u_2 = (-2,-1,2) u_3 = (-1,0,1)

Find a weighted Euclidean inner product on \(R^{n}\) such that the vectors \(v_{1}=(1,0,0, \ldots, 0)\) \(v_{2}=(0, \sqrt{2}, 0, \ldots, 0)\) \(v_{3}=(0,0, \sqrt{3}, \ldots, 0)\) \(\vdots\) \(v_{n}=(0,0,0, \ldots, \sqrt{n})\) form an orthonormal set. Equation Transcription: Text Transcription: R_n v_1 = (1,0,0, ldots, 0) v_2 = (0, sqrt 2, 0, ldots, 0) v_3 = (0,0, sqrt 3, ldots, 0) Vdots v_n = (0,0,0, ldots,sqrt n)

Is there a weighted Euclidean inner product on \(R^{2}\) for which the vectors \((1,2)\) and \((3,-1)\) form an orthonormal set? Justify your answer. Equation Transcription: Text Transcription: R^2 (1,2) (3,-1)

If \(u\) and \(v\) are vectors in an inner product space \(V\), then \(u,v\) and \(\mathbf{u}-\mathbf{v}\) can be regarded as sides of a "triangle" in \(V\) (see the accompanying figure). Prove that the law of cosines holds for any such triangle; that is, \(\|\mathbf{u}-\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}-2\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta\) where \(\theta\) is the angle between \(u\) and \(v\). Equation Transcription: Text Transcription: u v V u,v u-v |u-v|2 = |u|2 + |v|2 - 2|u| |v| cos theta

a. As shown in Figure \(3.2.6\), the vectors \((k, 0,0),(0, k, 0), \text { and }(0,0, k)\) form the edges of a cube in \(R^{3}\) with diagonal \((k, k, k)\). Similarly, the vectors \((k, 0,0, \ldots, 0),(0, k, 0, \ldots, 0), \ldots,(0,0,0, \ldots, k)\) can be regarded as edges of a "cube" in \(R^{n}\) with diagonal \((k, k, k, \ldots, k)\). Show that each of the above edges makes an angle of \(\theta\) with the diagonal, where \(\cos \theta=1 / \sqrt{n}\). b. (Calculus required) What happens to the angle \(\theta\) in part \((a)\) as the dimension of \(R^{n}\) approaches \(\infty\) ? Equation Transcription: Text Transcription: 3.2.6 (k,0,0), (0,k,0) and (0,0,k) R^3 (k,k,k) (k,0,0,ldots,0), (0,k,0,ldots,0), ldots , (0,0,0,ldots,k) R^n (k,k,k,ldots,k) cos theta =1/ sqrt n theta (a) infinity

Let \(u\) and \(v\) be vectors in an inner product space. a. Prove that \(\|\mathbf{u}\|=\|\mathbf{v}\|\) if and only if \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{u}-\mathbf{v}\) are orthogonal. b. Give a geometric interpretation of this result in \(R^{2}\) with the Euclidean inner product. Equation Transcription: Text Transcription: u v |u| = |v| u+v u-v R^2

Let \(u\) be a vector in an inner product space \(V\), and let \(\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\}\) be an orthonormal basis for \(V\). Show that if \(\alpha_{i}\) is the angle between \(\mathbf{u} \text { and } \mathbf{v}_{i}\), then \(\cos ^{2} \alpha_{1}+\cos ^{2} \alpha_{2}+\cdots+\cos ^{2} \alpha_{n}=1\) Equation Transcription: Text Transcription: u V {v_1, v_2, ldots , v_n} alpha_i u and v_i cos^2 alpha_1 + cos^2 alpha 2 + ldots + cos^2 alpha_n = 1

Prove: If \(\langle\mathbf{u}, \mathbf{v}\rangle_{1}\) and \(\langle\mathbf{u}, \mathbf{v}\rangle_{2}\) are two inner products on a vector space \(V\), then the quantity \(\langle\mathbf{u},\mathbf{v}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle_{1}+\langle\mathbf{u}, \mathbf{v}\rangle_{2}\) is also an inner product. Equation Transcription: Text Transcription: <u,v>_1 <u,v>_2 V <u,v> = <u,v>_1 + <u,v>_2

Prove Theorem \(6.2.5\). Equation Transcription: Text Transcription: 6.2.5

Prove: If \(A\) has linearly independent column vectors, and if \(b\) is orthogonal to the column space of \(A\), then the least squares solution of \(A x=b\) is \(x=0\). Equation Transcription: Text Transcription: A b Ax = b X = 0

Is there any value of \(s\) for which \(x_{1}=1\) and \(x_{2}=2\) is the least squares solution of the following linear system? \(x_{1}-x_{2}=1\) \(2 x_{1}+3 x_{2}=1\) \(4 x_{1}+5 x_{2}=s\) Explain your reasoning. Equation Transcription: Text Transcription: s x_1 = 1 x_2 = 2 x_1 - x_2 =1 2x_1 + 3x_2 = 1 4x_1 + 5x_2 = s

Show that if \(p\) and \(q\) are distinct positive integers, then the functions \(f(x)=\sin p x\) and \(g(x)=\sin q x\) are orthogonal with respect to the inner product \(\langle f, g\rangle=\int_{0}^{2 \pi} f(x) g(x) d x\) Equation Transcription: Text Transcription: p q f(x)=sin px g(x)=sin qx <f,g> = integral_0 ^2 f(x)g(x)dx

Show that if \(p\) and \(q\) are positive integers, then the functions \(f(x)=\operatorname{cosp} x\) and \(g(x)=\sin q x\) are orthogonal with respect to the inner product \(\langle f, g\rangle=\int_{0}^{2 \pi} f(x) g(x) d x\) Equation Transcription: Text Transcription: p q f(x)=cospx g(x)=sinqx <f,g> = integral_0 ^2 f(x)g(x)dx

Let \(W\) be the intersection of the planes \(x+y+z=0 \text { and } x-y+z=0\) in \(R^{3}\) . Find an equation for \(W^{\perp}\). Equation Transcription: Text Transcription: W x+y+z=0 and x-y+z=0 R^3 W^perp

Prove that if \(a d-b c \neq 0\) then the matrix \(A=\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\) has a unique \(QR\) -decomposition \(A=Q R\), where \(Q=\frac{1}{\sqrt{a^{2}+c^{2}}}\left[\begin{array}{lr}a & -c \\c & a\end{array}\right]\) \(R=\frac{1}{\sqrt{a^{2}+c^{2}}}\left[\begin{array}{cc}a^{2}+c^{2} & a b+c d \\0 & a d-b c\end{array}\right]\) Equation Transcription: Text Transcription: ad - bc neq 0 A=[a & b \\c & d] QR A = QR Q = 1/ sqrt a^2 + c^2 [a & -c \\c & a] R = 1/ sqrt a^2 + c^2 [a^{2}+c^{2} & a b+c d \\0 & a d-b c]