3 Geometry of matrices Matrix algebra is extremely useful in the formulation and solution of sets of linear equations. At the same time, the algebraic results have a geometrical basis that is very helpful in understanding the calculation. It is helpful to digress to a geometric treatment of matrices and vectors 3.1 Vector Spaces 3.1.1 Euclidean Vector Space Perhaps the most elementary vector are the Euclidean vector space R", n 1, 2,.... For simplicity, let us consider first R2. Non zero vector in R2 can be represented geometrically by directed line segments. Given a nonzero vector we can associate it with the line segment in the plane from(0, 0) to (1, 12). If we equate line segment that have the same length and direction, x can be represented by any line segment from(a, b)to(a+a1, b+r2). For example, the vector x I in R2 could be represented by the directed line segment from(2,2)to(4,3), or fron(-1,-1)to(1,0) We can think of the euclidean length of a vector x= as the length of any directed line segment representing x. The length of the segment from(0, 0) Two basic operations are defined for vectors, scalar multiplication and ad dition. The geometric representation will help us to visualize how the operation of scalar multiplication and addition work in R2 (1). Scalar multiplication: For each vector x 1 d scalar a. the product ax is defined by a t1 For example, the set of all possible scalar multiple of x is the line through 0 and x. Any scalar multiple of x is a segment of this line Example
3 Geometry of Matrices Matrix algebra is extremely useful in the formulation and solution of sets of linear equations. At the same time, the algebraic results have a geometrical basis that is very helpful in understanding the calculation. It is helpful to digress to a geometric treatment of matrices and vectors. 3.1 Vector Spaces 3.1.1 Euclidean Vector Space Perhaps the most elementary vector are the Euclidean vector space R n , n = 1, 2, .... For simplicity, let us consider first R 2 . Non zero vector in R 2 can be represented geometrically by directed line segments. Given a nonzero vector x = x1 x2 , we can associate it with the line segment in the plane from (0, 0) to (x1, x2). If we equate line segment that have the same length and direction, x can be represented by any line segment from (a, b) to (a + x1, b + x2). For example, the vector x = 2 1 in R 2 could be represented by the directed line segment from (2, 2) to (4, 3), or from (−1, −1) to (1, 0). We can think of the Euclidean length of a vector x = x1 x2 as the length of any directed line segment representing x. The length of the segment from (0, 0) to (x1, x2) is p x 2 1 + x 2 2 . Two basic operations are defined for vectors, scalar multiplication and addition. The geometric representation will help us to visualize how the operation of scalar multiplication and addition work in R 2 . (1). Scalar multiplication: For each vector x = x1 x2 and each scalar α, the product αx is defined by αx = αx1 αx2 . For example, the set of all possible scalar multiple of x is the line through 0 and x. Any scalar multiple of x is a segment of this line. Example: 6
2 The vector x*(= 2x) is in the same direction as x, but its length is two that of x. The vector x**(=-x) has half of length as x but its point in in the opposite direction (2). Addition: The sum of two vectors a and b is a third vector whose coordinates are the sums of the corresponding coordinates of a and b. For example +b Geometrically, c is obtained by moving in the distance and direction defined by b from the tip of a or, because addition is commutative from the tip of b in the distance and direction of a In a similar manner, vectors in R can be represented by directed line segments in a 3-space. Vector in Rn can be views as the coordinates of a point in a n- dimensional space or as the definition of the line segment connecting the origin and this point In general, scalar multiplication and addition in Rn are defined by and + 2+y2 for any x and y e Rn and any scalar a
x = 1 2 x ∗ = 2x = 2 4 x ∗∗ = − 1 2 x = − 1 2 −1 . The vector x ∗ (= 2x) is in the same direction as x, but its length is two times that of x. The vector x ∗∗(= − 1 2 x) has half of length as x but its point in the opposite direction. (2). Addition: The sum of two vectors a and b is a third vector whose coordinates are the sums of the corresponding coordinates of a and b. For example , c = a + b = 1 2 + 2 1 = 3 3 . Geometrically, c is obtained by moving in the distance and direction defined by b from the tip of a or, because addition is commutative, from the tip of b in the distance and direction of a. In a similar manner, vectors in R 3 can be represented by directed line segments in a 3-space. Vector in R n can be views as the coordinates of a point in a ndimensional space or as the definition of the line segment connecting the origin and this point. In general, scalar multiplication and addition in R n are defined by αx = αx1 αx2 . . . αxn and x + y = x1 + y1 x2 + y2 . . . xn + yn for any x and y ∈ R n and any scalar α. 7
3.1.2 Vector Space Axiom Definiti Let v be a set on which the operations of addition and scalar multiplication are (1).Ifx,y∈, then x+y∈, (2).Ifx∈ V and a is a scalar, then ax∈v The set v together with the operations of addition and scalar multiplication said to form a vector space if the following axioms are satisfied (a) x+y=y+x fo (b).(x+y)+z=x+(y+z) (c). There exist an element 0 in V such that x+0=x for each E V (d). For each x∈ 0 (e). a(x+y)=ax+ay for each real number a and any x and y in v (f).(a+ B)x=ax+ Bx for any real number a and B and any xE V (g).(aB)x=a(Bx)for any real number a and B and any xE V (h).1 The two-dimensional plane is the set of all vectors with two real-valued coordi nates. We label this set R2. It has two important properties (1).R is closed under scalar multiplication; every scalar multiple of a vector in the plane is also in the plane (2). R is closed under addition; the sum of any two vectors is always a vector in the plane 3.2 Linear Combination of vectors and basis vectors Definition A set of vectors in a vector is a basis for that vector space if any vector in the vector space can be written as a linear combination of them Example Any pair of two dimensional vectors that point in different directions will form
3.1.2 Vector Space Axiom Definition: Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that (1). If x, y ∈ V, then x + y ∈ V, (2). If x ∈ V and α is a scalar, then αx ∈ V. The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied: (a). x + y = y + x for any x and y in V. (b). (x + y) + z = x + (y + z). (c). There exist an element 0 in V such that x + 0 = x for each x ∈ V. (d). For each x ∈ V, there exist an element −x ∈ V such that x + (−x) = 0. (e). α(x + y) = αx + αy for each real number α and any x and y in V. (f). (α + β)x = αx + βx for any real number α and β and any x ∈ V. (g). (αβ)x = α(βx) for any real number α and β and any x ∈ V. (h). 1 · x = x for all x ∈ V. Example: The two-dimensional plane is the set of all vectors with two real-valued coordinates. We label this set R 2 . It has two important properties. (1). R 2 is closed under scalar multiplication; every scalar multiple of a vector in the plane is also in the plane. (2). R 2 is closed under addition; the sum of any two vectors is always a vector in the plane. 3.2 Linear Combination of Vectors and Basis Vectors Definition: A set of vectors in a vector is a basis for that vector space if any vector in the vector space can be written as a linear combination of them. Example: Any pair of two dimensional vectors that point in different directions will form 8
a basis for r2 Consider an arbitrary set of vectors in R, a, b, and c. If a and b are a basis, we can find numbers a and a% such that c=ara+a2b. Let b b2 C2 Then C1=a1a1+a2b1, C2=a1a2+ The solutions to this pair of equations are a1 C2-a2C1 b2- b1a2 This gives a unique solution unless(a1b2-b1a2)=0. If(a1b2-b1a2)=0, then a1/a2=61/b2, which means that b is just a multiple of a. This returns us to our original condition, that a and b point in different direction. The implication is that if a and b are any pair of vectors for which the denominator in o is not zero, then any other vector c can be formed as a unique linear combination of and b. The basis of a vector space is not unique, since any set of vectors that satisfy the definition will do. But for any particular basis, there is only one linear combination of them that will produce another particular vector in the vector space. 3.3 Linear Dependence As the preceding should suggest, k vectors are required to form a basis for R However it is not every set of k vectors will suffices. As we see, to form a basis we require that this k vectors to be linearly independent Definition: A sets of vectors is linearly dependent if any one of the vectors in the set can be
a basis for R 2 . Proof: Consider an arbitrary set of vectors in R 2 , a, b, and c. If a and b are a basis, we can find numbers α1 and α2 such that c = α1a + α2b. Let a = a1 a2 , b = b1 b2 , and c = c1 c2 . Then c1 = α1a1 + α2b1, c2 = α1a2 + α2b2. The solutions to this pair of equations are α1 = b2c1 − b1c2 a1b2 − b1a2 , (1) α2 = a1c2 − a2c1 a1b2 − b1a2 . (2) This gives a unique solution unless (a1b2 − b1a2) = 0. If (a1b2 − b1a2) = 0, then a1/a2 = b1/b2, which means that b is just a multiple of a. This returns us to our original condition , that a and b point in different direction. The implication is that if a and b are any pair of vectors for which the denominator in () is not zero, then any other vector c can be formed as a unique linear combination of a and b. The basis of a vector space is not unique, since any set of vectors that satisfy the definition will do. But for any particular basis, there is only one linear combination of them that will produce another particular vector in the vector space. 3.3 Linear Dependence As the preceding should suggest, k vectors are required to form a basis for R k . However it is not every set of k vectors will suffices. As we see, to form a basis we require that this k vectors to be linearly independent. Definition: A sets of vectors is linearly dependent if any one of the vectors in the set can be 9
written as a linear combination of the others Definition The vector V1, V2, . Vn in a vector space V are said to be linearly independent if and only if the solution to C1V1+C2V2+ The vector d 2 are linear independent, since if C +c2 0 C1+2c2=0 and the only solution to this system is 3.4 Subspace Definition The set of all linear combinations of a set of vectors is the vector space that is spanned by those vectors Example R=Spn(v1…,k) for a basis(v1,…,vk) We now consider what happens to the vector space that is spanned by linearly dependent vectors
written as a linear combination of the others. Definition: The vector v1, v2, ..., vn in a vector space V are said to be linearly independent if and only if the solution to c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0. Example: The vector 1 1 and 1 2 are linear independent, since if c1 1 1 + c2 1 2 = 0 0 , then c1 + c2 = 0 c1 + 2c2 = 0 and the only solution to this system is c1 = c2 = 0. 3.4 Subspace Definition: The set of all linear combinations of a set of vectors is the vector space that is spanned by those vectors. Example: R k = Span(v1, ..., vk) for a basis (v1, ..., vk). We now consider what happens to the vector space that is spanned by linearly dependent vectors. 10