Review - Ch. 5 & 7: Inner Product Spaces & Eigenvalues and Eigenvectors

3, √14, √5

LARLINALG8 5.1.008. SHOW YOUR WORK Find ||u||, ||v||, and ||u + v||. (a) ||u|| __________ (b) ||v|| __________ (c) ||u + v|| __________

(0, 1, -1/2, 3/2), (0, -4, 2, -6)

LARLINALG8 5.1.017. Given the vector v = (0, 2, −1, 3), find u according to the following parameters. (a) u has the same direction as v and one-half its length u = __________ (b) u has the direction opposite that of v and twice its length u = __________

√10

LARLINALG8 5.1.022. SHOW YOUR WORK Find the distance between u and v. d(u, v) = __________

8, 34, 61, (40, 0, 0, -24), 40

LARLINALG8 5.1.026. Find u · v, v · v, IIuII² , (u · v)v, and u · (5v). (a) u · v __________ (b) v · v __________ (c) ||u||² __________ (d) (u · v)v __________ (e) u · (5v) __________

-8

LARLINALG8 5.1.028. Find (3u − 2v) · (2u − 3v), given that u · u = 9, u · v = 8, and v · v = 7.

π/4

LARLINALG8 5.1.045. Find the angle 𝜃 between the vectors. 𝜃 = __________ radians

parallel

LARLINALG8 5.1.054. Determine whether u and v are orthogonal, parallel, or neither.

0, yes, 29, 25, (1, -2, -7), 54, yes

LARLINALG8 5.1.065.SBS. Verify the Pythagorean Theorem for the vectors u and v. STEP 1: Compute u · v. __________ Are u and v orthogonal? __________ STEP 2: Compute ||u||² and ||v||². ||u||² = __________ ||v||² = __________ STEP 3: Compute u + v and ||u + v||² . u + v = __________ ||u + v||² = __________ STEP 4: Is the statement ||u + v||² = ||u||² + ||v||² true? __________

(-21/29, 20/29), (21/29, -20/29)

LARLINALG8 5.1.078. Let v = (v₁, v₂) be a vector in R². Show that (v₂, −v₁) is orthogonal to v, and use this fact to find two unit vectors orthogonal to the given vector. __________ (smaller first component) __________ (larger first component)

25, 17, (-15/17, -8/17), (-4/5, -3/5), -420, 625, 289

LARLINALG8 5.1.503.XP. Use a graphing utility or computer software program with vector capabilities to find the following. (Enter each vector as a comma-separated list of its components.) (a) Norm of u and v ||u|| = __________ ||v|| = __________ (b) A unit vector in the direction of v __________ (c) A unit vector in the direction opposite that of u __________ (d) u · v __________ (e) u · u __________ (f) v · v __________

satisfies <u, v> = <v, u>, satisfies <u, v + w> = <u, v + u, w>, satisfies c<u, v> = <cu, v>, satisfies <v, v> ≥ 0, and <v, v> = 0 if and only if v = 0

LARLINALG8 5.2.005. Determine if the function defines an inner product on R³, where u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃). (Select all that apply.) - satisfies <u, v> = <v, u> - does not satisfy <u, v> = <v, u> - satisfies <u, v + w> = <u, v + u, w> - does not satisfy <u, v + w> = <u, v + u, w> - satisfies c<u, v> = <cu, v> - does not satisfy c<u, v> = <cu, v> - satisfies <v, v> ≥ 0, and <v, v> = 0 if and only if v = 0 - does not satisfy <v, v> ≥ 0, and <v, v> = 0 if and only if v = 0

0, √192, √627, √819

LARLINALG8 5.2.023. Find <u, v>, ||u||, ||v||, and d(u, v) for the given inner product defined on R^n. u = (8, 0, −8), v = (8, 9, 16), <u, v> = 2u₁v₁ + 3u₂v₂ + u₃v₃ a) <u, v> = __________ (b) ||u|| = __________ (c) ||v|| = __________ (d) d(u, v) = __________

-18, √52, √10, √98

LARLINALG8 5.2.029. Find <A, B>, ||A||, ||B||, and d(A, B) for the matrices in M₂,₂ using the inner product <A, B> = 2a₁₁b₁₁ + a₂₁b₂₁ + a₁₂b₁₂ + 2a₂₂b₂₂

-4, √26, √2, 6

LARLINALG8 5.2.035. Use the inner product <p, q> = a₀b₀ + a₁b₁ + a₂b₂ to find <p, q>, ||p||, ||q||, and d(p, q) for the polynomials in P₂. (a) <f, g> = __________ (b) ||f|| = __________ (c) ||g|| = __________ (d) d(f, g) = __________

4, √2, (4/5)√35, (√410)/5

LARLINALG8 5.2.039. Use the functions f and g in C[−1, 1] to find <f, g>, ||f||, ||g||, and d(f, g) for the inner product (a) <f, g> = __________ (b) ||f|| = __________ (c) ||g|| = __________ (d) d(f, g) = __________

π/2

LARLINALG8 5.2.047. Find the angle 𝜃 between the vectors. u = (1, 1, 1), v = (8, −8, 8), <u, v> = u₁v₁ + 2u₂v₂ + u₃v₃ 𝜃 = __________ radians

0, 3, √26, yes, 5x² + 3x + 1, √35, yes

LARLINALG8 5.2.057.SBS. Verify the Cauchy-Schwarz Inequality and the triangle inequality for the given vectors and inner product. p(x) = 3x, q(x) = 5x² + 1 <p, q> = a₀b₀ + a₁b₁ + a₂b₂ (a) Verify the Cauchy-Schwarz Inequality. STEP 1: Compute <p, q>. __________ STEP 2: Compute ||p||, and ||q||. ||p|| = __________ ||q|| = __________ STEP 3: Is |<p, q>| ≤ ||p|| ||q||? __________ (b) Verify the triangle inequality. STEP 1: Compute p + q and ||p + q||. p + q = __________ ||p + q|| = __________ STEP 2: Is ||p + q|| ≤ ||p|| + ||q||? __________

yes

LARLINALG8 5.2.066. Are f and g orthogonal in the inner product space C[a, b] with the inner product?

(4, -4, 0), (24/35, -8/7, 8/35)

LARLINALG8 5.2.073. Find projvu and projuv. Use the Euclidean inner product. (a) projvu __________ (b) projuv __________

(6e^x)/(e² - 1)

LARLINALG8 5.2.079. Find the orthogonal projection of f onto g. Use the inner product in C[a, b] projgf = __________

x is an eigenvector, x is not an eigenvector, x is an eigenvector, x is an eigenvector

LARLINALG8 7.1.011. Determine whether x is an eigenvector of A. (Type either "x is an eigenvector." or "x is not an eigenvector.") (a) x = (6, −2, 3) __________ (b) x = (6, 0, 3) __________ (c) x = (20, 4, −6) __________ (d) x = (0, 1, 1) __________

λ³ - 8λ² + 4λ + 48 = 0, (-2, 4, 6), <3/2, 1, 0>, <-3/2, 3, 1>, <-1/2, 1, 0>

LARLINALG8 7.1.025. Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. (a) the characteristic equation __________ (b) the eigenvalues (Enter your answers from smallest to largest.) (𝜆₁, 𝜆₂, 𝜆₃) = __________ a basis for each of the corresponding eigenspaces x₁ = __________ x₂ = __________ x₃ = __________

2, 3, 1

LARLINALG8 7.1.041. Find the eigenvalues of the triangular or diagonal matrix. (Enter your answers as a comma-separated list.) 𝜆 = __________

yes, yes

LARLINALG8 7.1.501.XP. Determine whether 𝜆i is an eigenvalue of A with the corresponding eigenvector xi. (a) 𝜆₁ = 2, x₁ = (1, 0) __________ (b) 𝜆₂ = -8, x₂ = (0, 1) __________

[4, 0, 0; 0, 3, 0; 0, 0, -5], (4, 3, -5)

LARLINALG8 7.2.005. (a) Verify that A is diagonalizable by computing P^−1AP. P^−1AP = __________ (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n × n matrices, then they have the same eigenvalues. (𝜆₁, 𝜆₂, 𝜆₃) = ___________

[-1, 7/2, 1; 1, -2, 0; 1, 1, 0], [1, 0, 0; 0, 4, 0; 0, 0, 2]

LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix P such that P^−1AP is diagonal. (If not possible, enter IMPOSSIBLE.) P = __________ Verify that P^−1AP is a diagonal matrix with the eigenvalues on the main diagonal. P^−1AP = __________

(3, 4), (1, 0, 0), (-1, 4, 1), three

LARLINALG8 7.2.019.SBS. Show that the matrix is not diagonalizable. STEP 1: Use the fact that the matrix is triangular to write down the eigenvalues. (Enter your answers from smallest to largest.) (𝜆₁, 𝜆₂) = __________ STEP 2:Find the eigenvectors x1 and x2 corresponding to 𝜆1 and 𝜆2, respectively. x₁ = __________ x₂ = __________ STEP 3:Since the matrix does not have -[--Select--- one, two, or three] linearly independent eigenvectors, you can conclude that the matrix is not diagonalizable. __________

0, 8, yes

LARLINALG8 7.2.023. Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.) Sufficient Condition for Diagonalization If an n × n matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable. Find the eigenvalues. (Enter your answers as a comma-separated list.) 𝜆 = __________ Is there a sufficient number to guarantee that the matrix is diagonalizable? __________

The matrix is not diagonalizable because it only has one linearly independent eigenvector.

LARLINALG8 7.2.049. Show that the matrix is not diagonalizable. (Select one of the below) a. The matrix is not diagonalizable because k is not an eigenvalue. b. The matrix is not diagonalizable because [k; 6] is not an eigenvector. c. The matrix is not diagonalizable because it only has one linearly independent eigenvector. d. The matrix is not diagonalizable because it only has one distinct eigenvalue.

[3, 1; 1, 1], [-1/2, 0; 0, 1/2]

LARLINALG8 7.2.501.XP. For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal. (If not possible, enter IMPOSSIBLE.) P = __________ Verify that P^−1AP is a diagonal matrix with the eigenvalues on the main diagonal. P^−1AP = __________