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MATH 4389 Final: Linear_Algebra(part_2)

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University of Houston
MATH 4389

Linear Algebra and Matrix Theory Part 2 - Vector Spaces 1. References (1) S. Friedberg, A. Insel and L. Spence, Linear Algebra, Prentice-Hall. (2) M. Golubitsky and M. Dellnitz, Linear Algebra and Differential Equa- tions Using Matlab, Brooks-Cole. (3) K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall. (4) P. Lancaster and M. Tismenetsky, The Theory of Matrices, Aca- demic Press. 2. Definition and Examples Briefly, a vector space consists of a set of objects called vectors along with a set of objects called scalars. The vector space axioms concern the algebraic relationships among the vectors and scalars. They may be found in any of the above references. Informally, they are as follows. Vectors can be added to form new vectors. Vectors can also be multiplied by scalars to form new vectors. There is one vector, called the zero vector, that acts as an identity element with respect to addition of vectors. Each vector has a negative associated with it. The sum of a vector and its negative is the zero vector. Addition of vectors is associative and commutative. In summary, the set of vectors with the operation of vector addition is an abelian group. 1 2 The set of scalars is an algebraic field, such as the field of real numbers or the field of complex numbers. Multiplication of scalars by vectors distributes over addition of vectors and also over addition of scalars. The product of the unit element 1 of the field with any vector is that vector. Finally, multiplication of vectors by scalars is associative. Generically, we denote the set of vectors by V and the field of scalars by F. It is common to say that V is a vector space over F. Individual vectors will be denoted by lower case Latin letters and individual scalars by lower case Greek letters. We use the same symbol + to denote both addition of scalars and addition of vectors. Multiplication of scalars by scalars or scalars by vectors is indicated simply by conjoining the symbols for the multiplicands. Example 2.1. Let F be either R or C and let V = F m×n . Two vectors (ma- trices) are added by adding corresponding entries and a matrix is multiplied by a scalar by multiplying each entry by that scalar. The zero matrix is the matrix all of whose entries are 0. The negative of a matrix is obtained by multiplying it by -1. If m = 1 the vectors of this space are called row vectors and if n = 1 they are called column vectors. Example 2.2. Two directed line segments in the Euclidean plane are equiv- alent if they have the same length and the same direction. Let V be the set of equivalence classes. If u and v are vectors (i.e., elements of V), choose representative line segments such that the representative of v begins where the representative of u ends. Then u + v is represented by the line segment 3 from the initial point of u to the terminal point of v. To multiply a vector u by a real number α, let αu be represented by a line segment whose length is |α| times the length of u. Let αu have the same direction as u if α > 0 and the opposite direction if α < 0. These are the familiar geometric vectors of elementary mathematics. We leave it to you to explain how the zero vector and negatives of vectors are defined and to verify the other properties of a vector space. Example 2.3. Let F = R and let V = C (0,1), the set of all functions u : (0,1) → R with at least k continuous derivatives on (0,1). Vector addition and scalar multiplication are defined pointwise: (u + v)(t) = u(t) + v(t) and (αu)(t) = αu(t) for each t ∈ (0,1). 3. Subspaces Let V be a vector space over F and let W be a nonempty subset of V. We say that W is a subspace of V if it is a vector space over F with the operations of vector addition and scalar multiplication inherited from V. This means that W is closed under vector addition and scalar multiplication and that it contains the zero vector of V. There is a simple test for when a subset of V is a subspace. Theorem 3.1. W is a subspace of V if and only if αu + βv ∈ W for all u,v ∈ W and all α,β ∈ F. Example 3.1. In Example 2.3, let W = C (0,1), where r > k. 4 Definition 3.1. Let V be a vector space over F and let S be a nonempty subset of V. The span of S is the subspace sp(S) = {α 1 1 ··· + α uk k ∈ N;α ,·1· ,α ∈ k;u ,···1,u ∈ Sk We also say that sp(S) is the subspace spanned by the elements of S. m×n 1×n Definition 3.2. Let A ∈ F . The row space of A is the subspace of F spanned by the rows of A. The column space of A is the subspace of F m×1 spanned by the columns of A. The null space of A is the set of all solutions x ∈ Fn×1 of the homogeneous system Ax = 0. There are short ways of denoting these subspaces. The row space is 1×m n×1 {yA|y ∈ F }, the column space is {Ax|x ∈ F }, and the null space is {x ∈ F n×1|Ax = 0}. Theorem 3.2. Row equivalent matrices have the same row space and null space. Subspaces can be combined to form new subspaces. One way is by taking their intersection. Another is by forming their sum, defined as Definition 3.3. Let U and W be subspaces of a vector space V. Their sum is U + W = {u + w|u ∈ U,w ∈ W}. Theorem 3.3. U ∩ W and U + W are subspaces of V. 5 Solved Problems: 1. Tell whether the following sets S are subspaces of V or not. If the answer is no, explain. (1) V = R 1×2, S = {(x 1x 2|x 1 0 or x = 2}. (2) V = F n×1 , S = {x|Ax = b}, where A ∈ F m×n and b ∈ F m×1 are given, b 6= 0. 2 (3) V = C (0,1), S =All real solutions y on (0,1) of the homogeneous 00 0 differential equation y − 2y + y = 0. 4×4 ∗ ∗ (4) V = C , F = C, S = {H ∈ V|H = H}, where H is the conjugate t of H . (5) Same V and S as in the preceding problem, but F = R. Solution: (1) Not a subspace because it is not closed under vector addition. For example, (1,0) and (0,1) are both in S but (1,1) is not. (2) Not a subspace because it is not closed under vector addition, not closed under scalar multiplication, does not contain the zero vector, and other reasons. (3) S is a subspace of V. (4) Not a subspace because it is not closed under scalar multiplication by elements of C. ∗ (5) S is a subspace. Incidentally, a complex matrix satisfying H = H is called hermitian. 6 2. Describe in the simplest possible terms the row space of the matrix   1 2 0 1    0 1 1 0      1 2 0 1 Solution: The reduced row echelon form has the same row space as the given matrix. It is    1 0 −2 1     0 1 1 0     0 0 0 0 The row space is the span of the rows of this matrix, i.e., the set of row vectors of the form (α1,2 ,−21 +2α 1α ) where 1 and2α are arbitrary elements of F. 3. Describe in simplest possible terms the null space of the same matrix. Solution: The reduced row echelon form has the same null space. A vector x = (1 2x 3x4,x ) is in the null space if and only if x1− 23 + 4 = 0 x2+ x3= 0 You may think of two of the varia1les,2say x and x , as having arbitrary values and the other two variables as being determined by them according 7 to the last set of equations. After substituting, the null space is the set of all vectors of the form (x1,2 ,−2 ,−1 − 2x ) where 1 and2x are arbitrary elements of F. 4. Describe in simplest terms the column space of the same matrix. Solution: A vector y is in the column space of A if and only if there is a solution of the linear system Ax = y. In the present case the augmented matrix is   1 2 0 1 | y 1     0 1 1 0 | y 2   1 2 0 1 | y 3 Now row-reduce the augmented matrix to get the coefficient part in row echelon form, with symbolic calculations in the last column. The result is   1 0 −2 1 | y − 2y  1 2   0 1 1 0 | y2      0 0 0 0 | 3 − 1 Obviously, this system has a solution 3f a1d only if y −y = 0. The column space is the set of all vectors y = (y ,y ,y ) with y = y . 1 2 3 3 1 Unsolved Problems: 8 1. Describe the row space, the null space and the column space of the following matrix.   1 2 1      1 0 1      1 1 1 2. Show that the intersection of two subspaces of a vector space is a sub- space. 3. Let S1and S2be nonempty subsets of a vector space V. Show that sp(S ) + sp(S ) = sp(S ∪ S ). 1 2 1 2 4. Let A ∈ Fm×n. Show that if y ∈ F is in the row space of A and x ∈ F×1 is in the null space of A, then yx = 0. 4. Linear Independence, Bases and Coordinates Definition 4.1. Vector1 v ,·m·in a vector space V over F are linearly dependent if there are sca1ars αm,··· ,α , not all zero1 1uch that α v + ··· +m mv= 0. Otherwise1 v ,·m·are linearly independent. An informal way of expressing linear dependence is to say that there is a non-trivial linear combination of the given vectors which is equal to the zero vector. Definition 4.2. A vector space V is finite-dimensional if there is a finite linearly independent set of vectors in V which spans V. Such a set of vectors is called a basis for V. 9 1×n Example 4.1. In F , let i denote the vector whose entries are all 0 except the i , which is 1. Then {e 1··· ,en} is a basis for F×n called the standard basis. In Fm×n , let i,jdenote the matrix all of whose entries are 0 except the i,j , which is 1. Then {E i,j≤ i ≤ m;1 ≤ j ≤ n} is a basis for Fm×n . There is never just one basis for a finite-dimensional vector space. If v ,··· ,v is a basis then so, for example, is 2v ,··· ,2v . However, we 1 m 1 m have the following theorem. Theorem 4.1. Any two bases for a finite-dimensional vector space have the same number of elements. The set {v ,··· ,v } is a basis for V if and 1 m only if for each vector u ∈ V there are unique scalars1α ,··· mα such that u = α 1 1 ··· + α m m Definition 4.3. The dimension of a finite-dimensional vector space is the number of elements in a basis. The dimension of V is denoted by dim(V). If dim(V) = m and v ,1·· ,v kre linearly independent vectors in V, then k ≤ m and if k < m we may extend this set of vectors to form a basis v1,··· ,k ,vk+1,··· ,m . Similarly, if1u ,···r,u spans V then r ≥ m and we may select a subset of m of the given vectors which forms a basis for V.
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