- 3.1: Let , , and . Compute (a) (b) (c) the distance between and (d) (e) ...
- 3.2: Repeat Exercise 1 for the vectors , , and .
- 3.3: Repeat parts (a)(d) of Exercise 1 for the vectors , , and
- 3.4: Repeat parts (a)(d) of Exercise 1 for the vectors , , and
- 3.5: In Exercises 56, determine whether the given set of vectors forms a...
- 3.6: In Exercises 56, determine whether the given set of vectors forms a...
- 3.7: (a) The set of all vectors in that are orthogonal to a nonzero vect...
- 3.8: Show that and are orthonormal vectors, and find a third vector for ...
- 3.9: True or False: If u and v are nonzero vectors such that , then u an...
- 3.10: True or False: If u is orthogonal to , then u is orthogonal to v an...
- 3.11: Consider the points , , and . Find the point S in whose first compo...
- 3.12: . Consider the points , , and . Find the point S in whose third com...
- 3.13: Using the points in Exercise 11, find the cosine of the angle betwe...
- 3.14: Using the points in Exercise 12, find the cosine of the angle betwe...
- 3.15: Find the distance between the point and the plane .
- 3.16: Show that the planes and are parallel, and find the distance betwee...
- 3.17: In Exercises 1722, find vector and parametric equations for the lin...
- 3.18: In Exercises 1722, find vector and parametric equations for the lin...
- 3.19: In Exercises 1722, find vector and parametric equations for the lin...
- 3.20: In Exercises 1722, find vector and parametric equations for the lin...
- 3.21: In Exercises 1722, find vector and parametric equations for the lin...
- 3.22: In Exercises 1722, find vector and parametric equations for the lin...
- 3.23: In Exercises 2325, find a point-normal equation for the given plane...
- 3.24: In Exercises 2325, find a point-normal equation for the given plane...
- 3.25: In Exercises 2325, find a point-normal equation for the given plane...
- 3.26: Suppose that and are two sets of vectors such that and are orthogon...
- 3.27: Prove that if two vectors u and v in are orthogonal to a nonzero ve...
- 3.28: Prove that if and only if u and v are parallel vectors.
- 3.29: The equation represents a line through the origin in if A and B are...
Solutions for Chapter 3: Euclidean Vector Spaces
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Every v in V is orthogonal to every w in W.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).