 2.1.1: Label the following statements as true or false. In each part, V an...
 2.1.2: For Exercises 2 through 6, prove that T is a linear transformation,...
 2.1.3: For Exercises 2 through 6, prove that T is a linear transformation,...
 2.1.4: For Exercises 2 through 6, prove that T is a linear transformation,...
 2.1.5: For Exercises 2 through 6, prove that T is a linear transformation,...
 2.1.6: For Exercises 2 through 6, prove that T is a linear transformation,...
 2.1.7: Prove properties 1, 2, 3, and 4 on page 65.
 2.1.8: Prove that the transformations in Examples 2 and 3 are linear.
 2.1.9: In this exercise, T: R2 > R2 is a function. For each of the followi...
 2.1.10: Suppose that T: R2 > R2 is linear, T(1,0) = (1,4), and T(l, 1) = (...
 2.1.11: Prove that there exists a linear transformation T: R2 > R3 such tha...
 2.1.12: Is there a linear transformation T: R3 R2 such that T(l, 0,3) = (1,...
 2.1.13: Let V and W be vector spaces, let T: V > W be linear, and let {w\,i...
 2.1.14: Let V and W be vector spaces and T: V > W be linear. (a) Prove that...
 2.1.15: Recall the definition of P(R) on page 10. Define T:P(JJ)P(Jl) by T...
 2.1.16: Let T: P(R) > P{R) be defined by T(/(x)) = f'(x). Recall that T is...
 2.1.17: Let V and W be finitedimensional vector spaces and T: V linear. W ...
 2.1.18: Let V and W be finitedimensional vector spaces and T: V linear. W ...
 2.1.19: Give an example of distinct linear transformations T and U such tha...
 2.1.20: Let V and W be vector spaces with subspaces Vi and Wi, respectively...
 2.1.21: Let V be the vector space of sequences described in Example 5 of Se...
 2.1.22: Let T: R3 R be linear. Show that there exist scalars a, b, and c su...
 2.1.23: Let T: R > R be linear. Describe geometrically the possibilities fo...
 2.1.24: The following definition is used in Exercises 24 27 and in Exercise...
 2.1.25: The following definition is used in Exercises 24 27 and in Exercise...
 2.1.26: The following definition is used in Exercises 24 27 and in Exercise...
 2.1.27: The following definition is used in Exercises 24 27 and in Exercise...
 2.1.28: The following definitions are used in Exercises 28 32. Definitions....
 2.1.29: The following definitions are used in Exercises 28 32. Definitions....
 2.1.30: The following definitions are used in Exercises 28 32. Definitions....
 2.1.31: The following definitions are used in Exercises 28 32. Definitions....
 2.1.32: The following definitions are used in Exercises 28 32. Definitions....
 2.1.33: Prove Theorem 2.2 for the case that ft is infinite, that is, R(T) =...
 2.1.34: Prove the following generalization of Theorem 2.6: Let V and W be v...
 2.1.35: Exercises 35 and 36 assume the definition of direct sum given in th...
 2.1.36: Exercises 35 and 36 assume the definition of direct sum given in th...
 2.1.37: A function T: V > W between vector spaces V and W is called additiv...
 2.1.38: Let T: C C be the function defined by T(z) = z. Prove that T is add...
 2.1.39: Prove that there is an additive function T: R > R (as defined in Ex...
 2.1.40: Let V be a vector space and W be a subspace of V. Define the mappin...
Solutions for Chapter 2.1: Linear Transformations, Null Spaces, and Ranges
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 2.1: Linear Transformations, Null Spaces, and Ranges
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4. Chapter 2.1: Linear Transformations, Null Spaces, and Ranges includes 40 full stepbystep solutions. Since 40 problems in chapter 2.1: Linear Transformations, Null Spaces, and Ranges have been answered, more than 11980 students have viewed full stepbystep solutions from this chapter. Linear Algebra was written by and is associated to the ISBN: 9780130084514.

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.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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).

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

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.

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!).

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.