- 3.1.1: In Exercises 12, find the components of the vector
- 3.1.2: In Exercises 12, find the components of the vector
- 3.1.3: In Exercises 34, find the components of the vector P1P2.(a) P1(3, 5...
- 3.1.4: In Exercises 34, find the components of the vector P1P2.(a) P1(6, 2...
- 3.1.5: (a) Find the terminal point of the vector that is equivalent to u =...
- 3.1.6: (a) Find the initial point of the vector that is equivalent to u = ...
- 3.1.7: Find an initial point P of a nonzero vector u = PQ with terminal po...
- 3.1.8: Find a terminal point Q of a nonzero vector u = PQ with initial poi...
- 3.1.9: Let u = (4, 1), v = (0, 5), and w = (3, 3). Find the components of ...
- 3.1.10: Let u = (3, 1, 2), v = (4, 0, 8), and w = (6, 1, 4). Find the compo...
- 3.1.11: Let u = (3, 2, 1, 0), v = (4, 7, 3, 2), and w = (5, 2, 8, 1). Find ...
- 3.1.12: Let u = (1, 2, 3, 5, 0), v = (0, 4, 1, 1, 2), and w = (7, 1, 4, 2, ...
- 3.1.13: Let u, v, and w be the vectors in Exercise 11. Find the components ...
- 3.1.14: Let u, v, and w be the vectors in Exercise 12. Find the components ...
- 3.1.15: Which of the following vectors in R6, if any, are parallel to u = (...
- 3.1.16: For what value(s) of t, if any, is the given vector parallel to u =...
- 3.1.17: Let u = (1, 1, 3, 5) and v = (2, 1, 0, 3). Find scalars a and b so ...
- 3.1.18: Let u = (2, 1, 0, 1, 1) and v = (2, 3, 1, 0, 2). Find scalars a and...
- 3.1.19: In Exercises 1920, find scalars c1, c2, and c3 for which the equati...
- 3.1.20: In Exercises 1920, find scalars c1, c2, and c3 for which the equati...
- 3.1.21: Show that there do not exist scalars c1, c2, and c3 such that c1(2,...
- 3.1.22: Show that there do not exist scalars c1, c2, and c3 such that c1(1,...
- 3.1.23: Let P be the point (2, 3, 2) and Q the point (7, 4, 1). (a) Find th...
- 3.1.24: In relation to the points P1 and P2 in Figure 3.1.12, what can you ...
- 3.1.25: In each part, find the components of the vector u + v + w. x y w v ...
- 3.1.26: Referring to the vectors pictured in Exercise 25, find the componen...
- 3.1.27: Let P be the point (1, 3, 7). If the point (4, 0, 6) is the midpoin...
- 3.1.28: If the sum of three vectors in R3 is zero, must they lie in the sam...
- 3.1.29: Consider the regular hexagon shown in the accompanying figure. (a) ...
- 3.1.30: What is the sum of all radial vectors of a regular n-sided polygon?...
- 3.1.31: Prove parts (a), (c), and (d) of Theorem 3.1.1.
- 3.1.32: Prove parts (e)(h) of Theorem 3.1.1
- 3.1.33: Prove parts (a)(c) of Theorem 3.1.2.
Solutions for Chapter 3.1: Vectors in 2-Space, 3-Space, and n-Space
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version | 11th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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 .
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Solvable system Ax = b.
The right side b is in the column space of A.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).