 4.1: TRUE OR FALSE? The space R2x3 is 5dimensional.
 4.2: TRUE OR FALSE?If / 1, is a basis of a linear space V, then any elem...
 4.3: TRUE OR FALSE?The space P\ is isomorphic to C .
 4.4: TRUE OR FALSE?If the kernel of a linear transformation T from P4 to...
 4.5: TRUE OR FALSE?If Wi and W2 are subspaces of a linear space V, then ...
 4.6: TRUE OR FALSE?If T is a linear transformation from P^ to R2x2, then...
 4.7: TRUE OR FALSE? The polynomials of degree less than 7 form a 7 dime...
 4.8: TRUE OR FALSE? The function T(f) = 3 /  4 /' from C to C is! linea...
 4.9: TRUE OR FALSE?The lower triangular 2 x 2 matrices form a subspace o...
 4.10: TRUE OR FALSE?The kernel of a linear transformation is a subspace o...
 4.11: TRUE OR FALSE?The linear transformation T(/) = f + f" from C to C i...
 4.12: TRUE OR FALSE?All linear transformations from P} to R2*2 are isomor...
 4.13: TRUE OR FALSE?If T is a linear transformation from V to V, then the...
 4.14: TRUE OR FALSE?The space of all upper triangular 4x4 matrices is iso...
 4.15: TRUE OR FALSE?Every polynomial of degree 3 can be expressed as a li...
 4.16: TRUE OR FALSE?If a linear space V can be spanned by 10 elements, th...
 4.17: TRUE OR FALSE?The function T(M) = det(M)fromR2x2toRisalinear transf...
 4.18: TRUE OR FALSE?There exists a 2 x 2 matrix A such that the space of ...
 4.19: TRUE OR FALSE?All bases of P3 contain at least one polynomial of de...
 4.20: TRUE OR FALSE?If must be an isomorT is an isomorphism, then T phism...
 4.21: TRUE OR FALSE?If the image of a linear transformation T from P to P...
 4.22: TRUE OR FALSE?If /h / 2 h is a basis of a linear space V, then f\, ...
 4.23: TRUE OR FALSE?If a, b, and c are distinct real numbers, then the po...
 4.24: TRUE OR FALSE?The linear transformation 7 (/(f)) = /(4f 3) from P t...
 4.25: TRUE OR FALSE?The linear transformation T(M) = R2x2 to R2x2 has ran...
 4.26: TRUE OR FALSE?If the matrix of a linear transformation T (with resp...
 4.27: TRUE OR FALSE?The kernel of the linear transformation T(/(f)) = /(f...
 4.28: TRUE OR FALSE?If S is any invertible 2 x 2 matrix, then the linear ...
 4.29: TRUE OR FALSE?There exists a 2 x 2 matrix A such that the space of ...
 4.30: TRUE OR FALSE?There exists a basis of R2x2 that consists of four in...
 4.31: TRUE OR FALSE?If W is a subspace of V, and if W is finite dimension...
 4.32: TRUE OR FALSE?There exists a linear transformation from R3x3 to R2x...
 4.33: TRUE OR FALSE?Every twodimensional subspace of R2x2 contains at le...
 4.34: TRUE OR FALSE?If = (/, g) and 23 = (/,/ + g) are two bases of a lin...
 4.35: TRUE OR FALSE?If the matrix of a linear transformation T with respe...
 4.36: TRUE OR FALSE?The linear transformation T(f) = f from Pn to Pn has ...
 4.37: TRUE OR FALSE?If the matrix of a linear transformation T (with resp...
 4.38: TRUE OR FALSE?There exists a subspace of R3x4 that is isomorphic to...
 4.39: TRUE OR FALSE?There exist two distinct subspaces W\ and W2 of R2x2 ...
 4.40: TRUE OR FALSE?There exists a linear transformation from P to P5 who...
 4.41: TRUE OR FALSE?If f\,..., / are polynomials such that the degree of ...
 4.42: TRUE OR FALSE?The transformation D(f) = / ' from C to C is an isomo...
 4.43: TRUE OR FALSE?If T is a linear transformation from P4 to W with im(...
 4.44: TRUE OR FALSE?The kernel of the linear transformation l T(f(t))= I ...
 4.45: TRUE OR FALSE?If T is a linear transformation from V to V, then {/ ...
 4.46: TRUE OR FALSE?If T is a linear transformation from P 6 to P6 that t...
 4.47: TRUE OR FALSE?There exist invertible 2 x 2 matrices P and Q such th...
 4.48: TRUE OR FALSE?There exists a linear transformation from P& to C who...
 4.49: TRUE OR FALSE? If f\ , / 2, /3 is a basis of a linear space V, and ...
 4.50: TRUE OR FALSE?There exists a twodimensional subspace of R2 x 2 who...
 4.51: TRUE OR FALSE?The space P\ \ is isomorphic to R3x4 .
 4.52: TRUE OR FALSE? If T is a linear transformation from V to W, and if ...
 4.53: TRUE OR FALSE?If T is a linear transformation from V to R2x2 with k...
 4.54: TRUE OR FALSE?The functiond /3r+4 w  t t f(x)dxfrom Ps to Ps is an...
 4.55: TRUE OR FALSE?Any 4dimensional linear space has infinitely many 3...
 4.56: TRUE OR FALSE?If the matrix of a linear transformation T (with resp...
 4.57: TRUE OR FALSE?If the image of a linear transformation T is infinite...
 4.58: TRUE OR FALSE?There exists a 2 x 2 matrix A such that the space of ...
 4.59: TRUE OR FALSE?If A, B, C, and D are noninvertible 2 x 2 matrices, t...
 4.60: TRUE OR FALSE?There exist two distinct 3dimensional subspaces W\ a...
 4.61: TRUE OR FALSE?If the elements /i,...,/ (where f\ ^ 0) are linearly ...
 4.62: TRUE OR FALSE?There exists a 3 x 3 matrix P such that the linear tr...
 4.63: TRUE OR FALSE?If f\, fi% / 3, / 4, fs are elements of a linear spac...
 4.64: TRUE OR FALSE?There exists a linear transformation T from P& to ffc...
 4.65: TRUE OR FALSE?If T is a linear transformation from V to W, and if b...
 4.66: TRUE OR FALSE? If the matrix of a linear transformation T (with res...
 4.67: TRUE OR FALSE?Every threedimensional subspace of IR2x2 contains at...
Solutions for Chapter 4: Linear Spaces
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 4: Linear Spaces
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 67 problems in chapter 4: Linear Spaces have been answered, more than 16744 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 4: Linear Spaces includes 67 full stepbystep solutions.

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

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.