 3.5.1: Questions 110 are about linear independence and linear dependence
 3.5.2: Questions 110 are about linear independence and linear dependence
 3.5.3: Questions 110 are about linear independence and linear dependence
 3.5.4: Questions 110 are about linear independence and linear dependence
 3.5.5: Questions 110 are about linear independence and linear dependence
 3.5.6: Questions 110 are about linear independence and linear dependence
 3.5.7: Questions 110 are about linear independence and linear dependence
 3.5.8: Questions 110 are about linear independence and linear dependence
 3.5.9: Questions 110 are about linear independence and linear dependence
 3.5.10: Questions 110 are about linear independence and linear dependence
 3.5.11: Questions 1115 are about the space spanned by a set of vectors. Ta...
 3.5.12: Questions 1115 are about the space spanned by a set of vectors. Ta...
 3.5.13: Questions 1115 are about the space spanned by a set of vectors. Ta...
 3.5.14: Questions 1115 are about the space spanned by a set of vectors. Ta...
 3.5.15: Questions 1525 are about the requirements for a basis
 3.5.16: Questions 1525 are about the requirements for a basis
 3.5.17: Questions 1525 are about the requirements for a basis
 3.5.18: Questions 1525 are about the requirements for a basis
 3.5.19: Questions 1525 are about the requirements for a basis
 3.5.20: Questions 1525 are about the requirements for a basis
 3.5.21: Questions 1525 are about the requirements for a basis
 3.5.22: Questions 1525 are about the requirements for a basis
 3.5.23: Questions 1525 are about the requirements for a basis
 3.5.24: Questions 1525 are about the requirements for a basis
 3.5.25: Questions 1525 are about the requirements for a basis
 3.5.26: Questions 2630 are about spaces where the "vectors" are matrices.
 3.5.27: Questions 2630 are about spaces where the "vectors" are matrices.
 3.5.28: Questions 2630 are about spaces where the "vectors" are matrices.
 3.5.29: Questions 2630 are about spaces where the "vectors" are matrices.
 3.5.30: Questions 2630 are about spaces where the "vectors" are matrices.
 3.5.31: Questions 3135 are about spaces where the "vectors" are functions.
 3.5.32: Questions 3135 are about spaces where the "vectors" are functions.
 3.5.33: Questions 3135 are about spaces where the "vectors" are functions.
 3.5.34: Questions 3135 are about spaces where the "vectors" are functions.
 3.5.35: Questions 3135 are about spaces where the "vectors" are functions.
 3.5.36: Find a basis for the space S of vectors (a, b, c, d) with a + c + ...
 3.5.37: If AS = SA for the shift matrix S, show that A must have this speci...
 3.5.38: Which of the following are bases for R 3? (a) (1,2,0) and (0, 1,1)...
 3.5.39: Suppose A is 5 by 4 with rank 4. Show that Ax = b has no solution w...
 3.5.40: (a) Find a basis for all solutions to d 4 y /dx 4 = y(x). (b) Find ...
 3.5.41: Write the 3 by 3 identity matrix as a combination of the other five...
 3.5.42: Choose x = (XI,X2,X3,X4) in R4. It has 24 rearrangements like (X2,X...
 3.5.43: Choose x = (XI,X2,X3,X4) in R4. It has 24 rearrangements like (X2,X...
 3.5.44: Mike Artin suggested a neat higherlevel proof of that dimension fo...
 3.5.45: Inside Rn, suppose dimension (V) + dimension (W) > n. Show that som...
 3.5.46: Suppose A is 10 by 10 and A2 = 0 (zero matrix). This means that the...
Solutions for Chapter 3.5: Independence, Basis and Dimension
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 3.5: Independence, Basis and Dimension
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.5: Independence, Basis and Dimension includes 46 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. Since 46 problems in chapter 3.5: Independence, Basis and Dimension have been answered, more than 12966 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Kirchhoff's Laws.
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.

lAII = 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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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 .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.