 3.6.1: (a) If a 7 by 9 matrix has rank 5, what are the dimensions of the f...
 3.6.2: Find bases and dimensions for the four subspaces associated with A ...
 3.6.3: Find a basis for each of the four subspaces associated with A:
 3.6.4: Construct a matrix with the required property or explain why this i...
 3.6.5: If V is the subspace spanned by (1,1,1) and (2,1,0), find a matrix ...
 3.6.6: Without elimination, find dimensions and bases for the four subspac...
 3.6.7: Suppose the 3 by 3 matrix A is invertible. Write down bases for the...
 3.6.8: What are the dimensions of the four subspaces for A, B, and C, if I...
 3.6.9: Which subspaces are the same for these matrices of different sizes?...
 3.6.10: If the entries of a 3 by 3 matrix are chosen randomly between 0 and...
 3.6.11: (Important) A is an m by n matrix of rank r. Suppose there are righ...
 3.6.12: Construct a matrix with (1,0,1) and (1,2,0) as a basis for its row ...
 3.6.13: True or false (with a reason or a counterexample): (a) If m = n the...
 3.6.14: Without computing A, find bases for its four fundamental subspaces:...
 3.6.15: If you exchange the first two rows of A, which of the four subspace...
 3.6.16: Explain why v = (1,0, 1) cannot be a row of A and also in the nUli...
 3.6.17: Describe the four subspaces of R3 associated with '" [0 1 0] A = 0 ...
 3.6.18: (Left nullspace) Add the extra column b and reduce A to echelon for...
 3.6.19: Following the method of 18, reduce A to echelon form and look at ze...
 3.6.20: (a) Check that the solutions to Ax = 0 are perpendicular to the row...
 3.6.21: Suppose A is the sum of two matrices of rank one: A = uv T + w z T....
 3.6.22: Construct A = uvT + wzT whose column space has basis (1,2,4), (2,2,...
 3.6.23: Without mUltiplying matrices, find bases for the row and column spa...
 3.6.24: Important) AT y = d is solvable when d is in which of the four subs...
 3.6.25: True or false (with a reason or a counterexample): (a) A and AT hav...
 3.6.26: (Rank of A B) If A B = C, the rows of C are combinations of the row...
 3.6.27: If a, b, c are given with a i= 0, how would you choose d so that [~...
 3.6.28: Find the ranks of the 8 by 8 checkerboard matrix B and the chess ma...
 3.6.29: Can tictactoe be completed (5 ones and 4 zeros in A) so that rank...
 3.6.30: Can tictactoe be completed (5 ones and 4 zeros in A) so that rank...
 3.6.31: M is the space of 3 by 3 matrices. Multiply every matrix X in M by ...
 3.6.32: Suppose the m by n matrices A and B have the same four subs paces. ...
Solutions for Chapter 3.6: Dimensions of the Four Subspaces
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 3.6: Dimensions of the Four Subspaces
Get Full SolutionsIntroduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Since 32 problems in chapter 3.6: Dimensions of the Four Subspaces have been answered, more than 11553 students have viewed full stepbystep solutions from this chapter. Chapter 3.6: Dimensions of the Four Subspaces includes 32 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Companion matrix.
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 nl  An).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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).

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.