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## Über dieses Buch

The present volume contains all the exercises and their solutions of Lang's' Linear Algebra. Solving problems being an essential part of the learning process, my goal is to provide those learning and teaching linear algebra with a large number of worked out exercises. Lang's textbook covers all the topics in linear algebra that are usually taught at the undergraduate level: vector spaces, matrices and linear maps including eigenvectors and eigenvalues, determinants, diagonalization of symmetric and hermitian maps, unitary maps and matrices, triangulation, Jordan canonical form, and convex sets. Therefore this solutions manual can be helpful to anyone learning or teaching linear algebra at the college level. As the understanding of the first chapters is essential to the comprehension of the later, more involved chapters, I encourage the reader to work through all of the problems of Chapters I, II, III and IV. Often earlier exercises are useful in solving later problems. (For example, Exercise 35, §3 of Chapter II shows that a strictly upper triangular matrix is nilpotent and this result is then used in Exercise 7, §1 of Chapter X.) To make the solutions concise, I have included only the necessary arguments; the reader may have to fill in the details to get complete proofs. Finally, I thank Serge Lang for giving me the opportunity to work on this solutions manual, and I also thank my brother Karim and Steve Miller for their helpful comments and their support.

## Inhaltsverzeichnis

### Chapter I. Vector Spaces

Abstract
Let V be a vector space. Using the properties VS 1 through VS 8,show that if c is a number then cO = O. SOLUTION. We have c0=(0+0)=c0+c0 but we also have c0=0+c0,Hence c0+c0=0+c0.
Rami Shakarchi

### Chapter II. Matrices

Abstract
Let $$A = \left( {\begin{array}{*{20}c} 1 & 2 & 3 \\ { - 1} & 0 & 2 \\ \end{array} } \right)$$ and $$B = \left( {\begin{array}{*{20}c} { - 1} & 5 & { - 2} \\ 2 & 2 & { - 1} \\ \end{array} } \right)$$. FindA+B, 3B, -2B, A+2B 2A-B, A-2B, B-A.
Rami Shakarchi

### Chapter III. Linear Mappings

Abstract
In Example 3, give Df as a function of x when fis the function:
$$f(x) = \sin x$$
(a)
$$f(x) = {e^x}$$
(b)
$$f(x) = \log x$$
(c)
Rami Shakarchi

### Chapter IV. Linear Maps and Matrices

Abstract
In each case, find the vector L A (X).
$$\begin{gathered} (a) A = \left( {\begin{array}{*{20}c} 2 & 1 \\ 1 & 0 \\ \end{array} } \right), X = \left( {\begin{array}{*{20}c} 3 \\ { - 1} \\ \end{array} } \right) (b) A = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right), X = \left( {\begin{array}{*{20}c} 5 \\ 1 \\ \end{array} } \right) \hfill \\ (c) A = \left( {\begin{array}{*{20}c} 1 & 1 \\ 0 & 1 \\ \end{array} } \right), X = \left( {\begin{array}{*{20}c} 4 \\ 1 \\ \end{array} } \right) (d) A = \left( {\begin{array}{*{20}c} 0 & 0 \\ 0 & 1 \\ \end{array} } \right),X = \left( {\begin{array}{*{20}c} 7 \\ { - 3} \\ \end{array} } \right) \hfill \\ \end{gathered}$$
SOLUTION. $$(a)\left( {\begin{array}{*{20}c} 5 \\ 3 \\ \end{array} } \right)(b)\left( {\begin{array}{*{20}c} 5 \\ 0 \\ \end{array} } \right)(c)\left( {\begin{array}{*{20}c} 5 \\ 1 \\ \end{array} } \right)(d)\left( {\begin{array}{*{20}c} 0 \\ { - 3} \\ \end{array} } \right).$$
Rami Shakarchi

### Chapter V. Scalar Products and Orthogonality

Abstract
Let V be a vector space with a scalar product. Show that (O,v)=0 for all v in V. SOLUTION. We have (O,v)=(v-v,v)=(v,v)-(v,v)=0.
Rami Shakarchi

### Chapter VI. Determinants

Abstract
Let c be a number and let A be a 3 x 3 matrix. Show that D(cA)=c3D(A). SOLUTION. See Excercise 2.
Rami Shakarchi

### Chapter VII. Symmetric, Hermitian, and Unitary Operators

Abstract
(a)
A matrix A is called skew-symmetric if t A= -A. Show that any matrix M can he expressed as a sum of a symmetric matrix and a skew-symmetric matrix one and that the latter expression is uniquely determined. [Hint:Let A = 1/2(M+1 M).]

(b)
Prove that if A is skew-symmetric,then A 2 is symmetric.

(c)
Let A be skew-symmetric. Show that Det(A) is 0 if A is an n x n matrix and n is odd.

Rami Shakarchi

### Chapter VIII. Eigenvectors and Eigenvalues

Abstract
Let a ∈ and a ≠0. Prove that the eigenvectors of the matrix
$$\left( {\begin{array}{*{20}c} 1 & a \\ 0 & 1 \\ \end{array} } \right)$$
generate a 1-dimensional space, and give a basis for this space.
Rami Shakarchi

### Chapter IX. Polynomials and Matrices

Abstract
Compute f(A) when f(t) = t3 - 2t + 1 and $$A = \left( {\begin{array}{*{20}c} { - 1} & 1 \\ 2 & 4 \\ \end{array} } \right)$$ .
Rami Shakarchi

### Chapter X. Triangulation of Matrices and Linear Maps

Abstract
Let A be an upper triangular matrix:
$$A = \left( {\begin{array}{*{20}c} \hfill {a_{11} } & \hfill {a_{12} } & \hfill \ldots & \hfill {a_{1n} } \\ \hfill 0 & \hfill {a_{22} } & \hfill \ldots & \hfill {a_{2n} } \\ \hfill : & \hfill : & \hfill {} & \hfill : \\ \hfill 0 & \hfill 0 & \hfill \ldots & \hfill {a_{nn} } \\ \end{array} } \right).$$
Viewing A as a linear map, what are the eigenvalues of A2, A3 in general Ar where r is an integer ≥1?
Rami Shakarchi

### Chapter XI. Polynomials and Primary Decomposition

Abstract
In each of the following cases,write f = qg + r with deg r < deg g.
(a) f(t)=t2-2t+1, g(t)=t-1
(b) f(t)=t3+t-1, g(t)=t2+1
(c) f(t)=t3+t, g(t)=t
(d) f(t)=t3-1, g(t)=t-1
SOLUTION
(a) f(t)=(t-1)g(t). (b) f(t)=tg(t)-1.
(c) f(t)=(t2+1)g(t). (d) f(t)=(t2+t+1)g(t).
Rami Shakarchi

### Chapter XII. Convex Sets

Abstract
1. Let A be a vector in RnLet F: RnR n be the translation F(X)=X+A.
Show that if S is convex in Rnthen F(S) is also convex.
Rami Shakarchi

### Backmatter

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