Introduction to Linear Algebra IV - Department of Mathematics

Introduction to Linear Algebra IV Jack Xin (Lecture) and J. Ernie Esser (Lab)

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Abstract Nullspace, nullity, dimension, linear transformation, determinant.

1

Nullspace, Nullity, Dimension

A homogeneous linear system is Ax = 0, the solution set has the property that any linear combination of two solutions remains a solution. A subset satisfying this property is called a subspace. The solution set is then called a null space N . A subspace on the plane is a line passing through the origin, for example W = {(x, 3x) ∈ R2 , ∀ x ∈ R1 }, while the set of vectors in the first quadrant: W1 = {(x, y) ∈ R2 : xy ≥ 0}, is not a subspace. The smallest subspace is (0, 0). A subset (w1 , w2 , · · · , wk ) of a subspace W is a basis if every vector of W is uniquely expressed as a linear combination of w1 , w2 , · · · , wk . The number k is the dimension of W , denoted by dim(W ). If we perform row reduction to transform a matrix A to H, we shall have a few number of columns with nonzero pivots and other pivots free columns. The number of pivots is the rank of A, the number of pivot free columns is called nullity. Clearly, rank(A) + nullity(A) = number of columns, which is the so called rank equation. Nullity is same as dim(N ). ∗

Department of Mathematics, UCI, Irvine, CA 92617.

1

(1.1)

Example 1: row reduction shows that  1 3 0 −1  0 −2 4 −2 A=  3 11 −4 −1 2 5 3 −4

  1 2   0 0  →  0 6  0 0

0 1 0 0

 0 2 26 0 −1 −8   1 −1 −4  0 0 0

rank(A)=3, nullity = 2, number of columns = 5. The rank equals the dimension of column space of A. Example 2: Let W = span(w1 , w2 , w3 , w4 ) ⊂ R3 , w1 =[1 -3 1], w2 =[-2 6 -2], w3 =[2 1 -4], w4 =[-1 10 -7]. Let us find the dimension and basis of subspace W . Put wi ’s into columns of a matrix A and rref:     1 −2 2 −1 1 −2 0 −3 1 10  →  0 0 1 1  A =  −3 6 1 −2 −4 −7 0 0 0 0 The 1st and 3rd columns are independent and span W , hence basis vectors are: w1 and w3 , and dim(W)=2. In fact, w2 = −2w1 , w4 = −3w1 + w3 . The solution structure of inhomogeneous linear system Ax = b has the form x = p + h, Ah = 0, Ap = b, h ∈ N , p is a particular solution.

2

Linear Transformation

A linear transformation (LT) is a function T : Rn → Rm such that (1) T (u + v) = T (u) + T (v), (2) T (c u) = cT (u), for any vectors u and v, any scalar c. Clearly, T (0) = 0. P Let e1 , e2 , · · · , en be standard basis of Rn , then x = ni=1 xi ei , so: n n X X Tx = T( xi ei ) = xi T (ei ) = Ax, i=1

i=1

where matrix A consists of columns T (e1 ), · · · , T (en ). Matrix A is the associated matrix of T . The range of T is the column space of A, the nullspace of T (kernel of T , ker(T )) is that of A. An example of linear transformation on the plane is rotation with its matrix:   cos θ − sin θ sin θ cos θ counter-clockwise rotation matrix of angle θ. 2

2.1

Lines, Planes, Flats

Solution of linear system Ax = b has geometric meaning as lines, planes or their extension called flats. A line is a translate of on dimensional subspace of Rn , obeying the vector equation: x = a + t d, where t is the parameter, a and d are given vectors. A k-flat in Rn is a translate of k-dimensional subspace of Rn . A single point is zero flat. A plane is a 2-flat. A hyperplane is (n-1)-dimensional flat. The k-flat equation is: x = a + t1 d1 + t2 d2 + · · · + tk dk , where t1 , · · · , tk are k-parameters. Example 3: Intersection of 3 hyperplanes in R4 satisfies the system of equations as follows (in augmented matrix form):   1 2 −2 1 3 | 1  2 5 −3 −1 2 | 2  −3 −8 6 −1 −5 | 1 which is rrefed into:



 1 0 0 −1 3 | 9  0 1 0 −1 −2 | −2  0 0 1 −2 −2 | 2

In view of two free columns, we see that the solution set is a 2-flat: x = [9 − 2 2 0 0] + t1 [1 1 2 1 0] + t2 [−3 2 2 0 1].

In general, if Ax = b is consistent, x ∈ Rn , b ∈ Rm , A ∈ Rm×n , the solution set is (n − r)-flat, r is rank(A).

3

Determinant

Determinant (det) of a number is the number itself, det of a 2 × 2 matrix is:   a11 a12 det = a11 a22 − a12 a21 . a21 a22 3

Let A be n × n matrix, Mij the (n − 1) × (n − 1) matrix by deleting the row and column containing aij . The cofactor of aij is the number Aij = (−1)i+j det(Mij ). Then if n > 1: det(A) = a11 A11 + a12 A12 + · · · + a1n A1n , cofactor expansion along the first row. The value of det is the same with cofactor expansion along any row or column. Det of an upper or lower triangular matrix is the product of diagonal entries. In Matlab, det of A is simply det(A). Properties of det are: 1. det(AB) = det(A) det(B). 2. det (E1 A) = - det (A), E1 is elementary matrix of exchanging two rows. 3. det(E3 A) = det (A), E3 is elementary matrix of adding a multiple on one row onto another. 4. det (E2 A) = a det(A), E2 is elementary matrix of multipling a row or column by scalar a. 5. A is invertible iff det(A)6= 0. Moreover, A−1 = adj(A)/det(A), where adj(A) is the adjoint matrix defined by (Aij )T . An interesting class of matrices is the integer matrices (matrices with integer entries). In applications of coding and decoding of message transmission, people are interested in integer matrices whose inverse are also integer matrices. Such matrices can be obtained by applying a sequence of E1 and E3 matrices (with integer multiples) to the identity matrix. The resulting matrix will have det = 1 or -1, and so have integer matrix as inverse as follows from property 5 above. An example is:   1 2 1  2 5 3  2 3 2 Find its inverse and check that it is also an integer matrix.

References [1] S. Leon, Linear Algebra with Applications, Pearson, Prentice-Hall, 2010. 4