solved problems in linear algebra pdf

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• Prof. Gilbert Strang

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Problems in Mathematics

Category: Linear Algebra

Linear Algebra Problems and Solutions.

Popular topics in Linear Algebra are

• Vector Space
• Linear Transformation
• Diagonalization

Gauss-Jordan Elimination   Inverse Matrix     Eigen Value   Caley-Hamilton Theorem   Caley-Hamilton Theorem

Linear Algebra

 by Yu · Published 04/30/2018

If the Nullity of a Linear Transformation is Zero, then Linearly Independent Vectors are Mapped to Linearly Independent Vectors

Problem 722.

Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that the nullity of $T$ is zero.

If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.

  Read solution

 by Yu · Published 04/22/2018

Find All Values of $x$ such that the Matrix is Invertible

Problem 721.

 by Yu · Published 04/15/2018

Find All Eigenvalues and Corresponding Eigenvectors for the $3\times 3$ matrix

Problem 720, find all values of $a$ which will guarantee that $a$ has eigenvalues 0, 3, and -3., problem 719.

Let $A$ be the matrix given by \[ A= \begin{bmatrix} -2 & 0 & 1 \\ -5 & 3 & a \\ 4 & -2 & -1 \end{bmatrix} \] for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$.

 by Yu · Published 04/03/2018

Compute the Determinant of a Magic Square

Problem 718.

Let \[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square.

Compute the determinant of $A$.

 by Yu · Published 03/25/2018

Are These Linear Transformations?

Problem 717.

Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x+y \\ 0 \end{bmatrix} ,\; S\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x+y \\ xy \end{bmatrix} . \] Determine whether $T$, $S$, and the composite $S\circ T$ are linear transformations.

Using Gram-Schmidt Orthogonalization, Find an Orthogonal Basis for the Span

Problem 716.

 by Yu · Published 03/21/2018

Normalize Lengths to Obtain an Orthonormal Basis

Problem 715.

Let \[ \mathbf{v}_{1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} ,\; \mathbf{v}_{2} = \begin{bmatrix} 1 \\ -1 \end{bmatrix} . \] Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$?

If not, then find an orthonormal basis for $V$.

 by Yu · Published 03/19/2018

Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices

Problem 714.

Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.

 by Yu · Published 03/07/2018

Determine Bases for Nullspaces $\calN(A)$ and $\calN(A^{T}A)$

Problem 713.

Determine bases for $\calN(A)$ and $\calN(A^{T}A)$ when \[ A= \begin{bmatrix} 1 & 2 & 1 \\ 1 & 1 & 3 \\ 0 & 0 & 0 \end{bmatrix} . \] Then, determine the ranks and nullities of the matrices $A$ and $A^{\trans}A$.

 by Yu · Published 03/05/2018

In which $\R^k$, are the Nullspace and Range Subspaces?

Problem 712.

Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.

 by Yu · Published 02/28/2018

Prove Vector Space Properties Using Vector Space Axioms

Problem 711.

Using the axiom of a vector space, prove the following properties. Let $V$ be a vector space over $\R$. Let $u, v, w\in V$.

(a) If $u+v=u+w$, then $v=w$.

(b) If $v+u=w+u$, then $v=w$.

(c) The zero vector $\mathbf{0}$ is unique.

(d) For each $v\in V$, the additive inverse $-v$ is unique.

(e) $0v=\mathbf{0}$ for every $v\in V$, where $0\in\R$ is the zero scalar.

(f) $a\mathbf{0}=\mathbf{0}$ for every scalar $a$.

(g) If $av=\mathbf{0}$, then $a=0$ or $v=\mathbf{0}$.

(h) $(-1)v=-v$.

The first two properties are called the cancellation law .

 by Yu · Published 02/26/2018

Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors

Problem 710.

Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$.

Find a Basis for the Subspace spanned by Five Vectors

Problem 709.

Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where \[ \mathbf{v}_{1}= \begin{bmatrix} 1 \\ 2 \\ 2 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{2}= \begin{bmatrix} 1 \\ 3 \\ 1 \\ 1 \end{bmatrix} ,\;\mathbf{v}_{3}= \begin{bmatrix} 1 \\ 5 \\ -1 \\ 5 \end{bmatrix} ,\;\mathbf{v}_{4}= \begin{bmatrix} 1 \\ 1 \\ 4 \\ -1 \end{bmatrix} ,\;\mathbf{v}_{5}= \begin{bmatrix} 2 \\ 7 \\ 0 \\ 2 \end{bmatrix} .\] Find a basis for the span $\Span(S)$.

How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix

Problem 708.

Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$.

(a) Find a basis for the nullspace of $A$.

(b) Find a basis for the row space of $A$.

(c) Find a basis for the range of $A$ that consists of column vectors of $A$.

(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.

Can We Reduce the Number of Vectors in a Spanning Set?

Problem 707.

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?

 by Yu · Published 02/25/2018

Does an Extra Vector Change the Span?

Problem 706.

Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set \[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.

 by Yu · Published 02/22/2018

Vector Space of Functions from a Set to a Vector Space

Problem 705.

(a) Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element?

(b) Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism.

(c) Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$.

(e) Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$?

(f) Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$.

 by Yu · Published 02/21/2018

Find a Basis for Nullspace, Row Space, and Range of a Matrix

Problem 704.

Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$. (a) Find a basis for the nullspace of $A$.

 by Yu · Published 02/15/2018

Describe the Range of the Matrix Using the Definition of the Range

Problem 703.

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Problems In Linear Algebra

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In preparing this book of problems the author attempted firstly, to give a sufficient number of exercises for developing skills in the solution of typical problems (for example, the computing of determinants with numerical elements, the solution of systems of linear equations with numerical coefficients, and the like), secondly, to provide problems that will help to clarify basic concepts and their interrelations (for example, the connection between the properties of matrices and those of quadratic forms, on the one hand and those of linear transformations, on the other), thirdly to provide for a set of problems that might supplement the course of lectures and help to expand the mathematical horizon of the student (instances are the properties of the Pfaffian of the skew-symmetric determinant, the properties of associated matrices, and so on). Compared with other problem book, this one has few new basic features. They include problems dealing with polynomial matrices (Sec. 13), linear transformations of affine and metric spaces (Secs. 18 and 19), and a supplement devoted to group rings, and fields. The problems of the supplement deal with the most elementary portions of the theory. Still and all, I think it can be used in pre-seminar discussions in the first and second years of study. Starred numbers indicated problems that have been worked out or provided with hints. Solutions are given for a small number of problems. The book was translated from the Russian by George Yankovsky and was first published by Mir Publishers in 1978. Contents Preface 5 Chapter I DETERMINANTS Sec. 1. Second and third-order determinants 9 Sec. 2. Permutations and substitutions 17 Sec. 3. Definition and elementary properties of determinants of any order 22 Sec. 4. Evaluating determinants with numerical elements 31 Sec. 5. Methods of computing determinants of the th order 33 Sec. 6. Monirs, cofactors and the Laplace theorem 65 Sec. 7. Multiplication of determinants 74 Sec. 8. Miscellaneous problems 86 Chapter II SYSTEMS OF LINEAR EQUATIONS Sec. 9. Systems of equation solved by the Cramer rule 95 Sec. 10. The rank of a matrix. The linear dependence of vectors and linear forms 105 Sec. 11. Systems of linear equations 115 Chapter III MATRICES AND QUADRATIC FORMS Sec. 12. Operations involving matrices 131 Sec. 13. Polynomial matrices 155 Sec. 14. Similar matrices, characteristic and minimal polynomials. Jordan and diagonal forms of a matrix. Functions of matrices. 166 Sec. 15. Quadratic forms 182 Chapter IV VECTOR SPACES AND THEIR LINEAR TRANSFORMATIONS Sec. 16. Affine vector spaces 195 Sec. 17. Euclidean and unitary vector spaces 205 Sec. 18. Linear transformations of arbitrary vector spaces 220 Sec. 19. Linear transformations of Euclidean and unitary vector spaces 236 Sec. 20. Groups 251 Sec. 21. Rings and fields 265 Sec. 22. Modules 275 Sec. 23. Linear spaces and linear transformations (appendices to Secs. 10 and 16 to 19) 280 Sec. 24. Linear, bilinear, and quadratic functions and forms (appendix to Sec. 15) 284 Sec. 25. Affine (or point-vector) spaces 288 Sec. 26. Tensor algebra 295 ANSWERS Chapter I. Determinants 312 Chapter II. Systems of linear equations 342 Chapter III. Matrices and quadratic forms 359 Chapter IV. Vector spaces and their linear transformations 397 Supplement 427 Index 449

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