 3.1.1: In Exercises 12, draw a coordinate system (as in Figure 3.1.10) and...
 3.1.2: In Exercises 12, draw a coordinate system (as in Figure 3.1.10) and...
 3.1.3: In Exercises 34, sketch the following vectors with the initial poin...
 3.1.4: In Exercises 34, sketch the following vectors with the initial poin...
 3.1.5: In Exercises 56, sketch the following vectors with the initial poin...
 3.1.6: In Exercises 56, sketch the following vectors with the initial poin...
 3.1.7: In Exercises 78, find the components of the vector .
 3.1.8: In Exercises 78, find the components of the vector .
 3.1.9: (a) Find the terminal point of the vector that is equivalent to and...
 3.1.10: (a) Find the initial point of the vector that is equivalent to and ...
 3.1.11: Find a nonzero vector u with terminal point such that (a) u has the...
 3.1.12: Find a nonzero vector u with initial point such that (a) u has the ...
 3.1.13: Let , , and . Find the components of (a) (b) (c) (d) (e) (f)
 3.1.14: Let , , and . Find the components of (a) (b) (c) (d) (e) (f)
 3.1.15: Let , , and . Find the components of (a) (b) (c) (d) (e) (f)
 3.1.16: Let u, v, and w be the vectors in Exercise 15. Find the vector x th...
 3.1.17: Let , , and . Find the components of (a) (b) (c) (d) (e) (f)
 3.1.18: Let and . Find the components of (a) (b) (c)
 3.1.19: Let and . Find the components of
 3.1.20: Let u, v, and w be the vectors in Exercise 18. Find the components ...
 3.1.21: Let u, v, and w be the vectors in Exercise 19. Find the components ...
 3.1.22: For what value(s) of t, if any, is the given vector parallel to ? (...
 3.1.23: Which of the following vectors in are parallel to ? (a) (b) (c)
 3.1.24: Let and Find scalars a and b so that
 3.1.25: Let and . Find scalars a and b so that
 3.1.26: Find all scalars , , and such that
 3.1.27: Find all scalars , , and such that
 3.1.28: Find all scalars , , and such that
 3.1.29: Let , , , and . Find scalars , , , and such that .
 3.1.30: Show that there do not exist scalars , , and such that
 3.1.31: Show that there do not exist scalars , , and such that
 3.1.32: Consider Figure 3.1.12. Discuss a geometric interpretation of the v...
 3.1.33: Let P be the point and Q the point . (a) Find the midpoint of the l...
 3.1.34: Let P be the point . If the point is the midpoint of the line segme...
 3.1.35: Prove parts (a), (c), and (d) of Theorem 3.1.1.
 3.1.36: Prove parts (e)(h) of Theorem 3.1.1.
 3.1.37: Prove parts (a)(c) of Theorem 3.1.2.
 3.1.a: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.b: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.c: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.d: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.e: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.f: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.g: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.h: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.i: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.j: In parts (a)(k) determine whether the statement is true or false, a...
 3.1.k: In parts (a)(k) determine whether the statement is true or false, a...
Solutions for Chapter 3.1: Vectors in 2Space, 3Space, and nSpace
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 3.1: Vectors in 2Space, 3Space, and nSpace
Get Full SolutionsSince 48 problems in chapter 3.1: Vectors in 2Space, 3Space, and nSpace have been answered, more than 13734 students have viewed full stepbystep solutions from this chapter. Chapter 3.1: Vectors in 2Space, 3Space, and nSpace includes 48 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051.

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.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).