Basis, Dimension, and OrthonormalityDate: 06/28/98 at 09:41:28 From: Naomi Subject: Linear algebra Hi! I have two questions: 1) How would you find a basis of the kernel, a basis of the image and determine the dimension of each for this matrix? The matrix is: 1 2 1 2 1 1 2 2 1 2 2 4 3 3 3 0 0 1 -1 -1 2) Are the following 3 vectors linearly dependent? How can you decide? 1 1 1 1 2 4 1 3 7 1 4 10 Date: 06/28/98 at 16:47:22 From: Doctor Anthony Subject: Re: Linear algebra Problem 1: The kernel is the vector subspace that maps into the zero vector. It represents the number of dimensions lost in the transformation. So: Dimension of range = dimension of domain - dimension of kernel The rank of the matrix is the number of linearly independent rows and columns, and the dimension of the range is the rank of the matrix. By row operations (which I shall leave for you to do) you partition the matrix into the form (known as the canonical form): [1 0 0 | a b] |0 1 0 | c d] |0 0 1 | e f] [-----------|------] [0 0 0 | 0 0] Assume that this is how it works out, and we can see that in this example the number of independent rows (or columns) is 3, and so the rank is 3. The basis of the kernel will be the vectors: [ a] [ b] | c| and | d] | e| | f| |-1| | 0| [ 0] [-1] Problem 2: If the vectors are linearly dependent, one of the vectors can be expressed as a linear combination of the other two. So, can you express vector 3 as a linear combination of vectors 1 and 2? If so, we have: 1 [1] [1] 4 = p|1| + q|2| 7 |1| |3| 10 [1] [4] From this we require: p + q = 1 p + 2q = 4 p + 3q = 7 p + 4q = 10 Solving the first two gives: q = 3, p = -2. Put this into the third equation: -2 + 9 = 7, which is correct, and into the fourth equation: -2 + 12 = 10, which is correct. So the third vector is (-2) times the first vector plus 3 times the second vector. It follows that the vectors are linearly dependent. - Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 07/03/98 at 12:37:19 From: Naomi Subject: Linear algebra Hi, I have another question. How would you perform the Gram-Schmidt process on: 2 3 5 0 4 6 0 0 7 Date: 07/03/98 at 18:57:50 From: Doctor Anthony Subject: Re: Linear algebra If M is the space R3 generated by {2,0,0}, {3,4,0}, {5,6,7}, we must find an orthonormal basis for M. The inner product (or dot product) of two vectors x = {x1,x2,x3} and y = {y1,y2,y3} is defined to be: x.y = x1 * y1 + x2 * y2 + x3 * y3 Note that if two vectors are orthonormal, their dot product is 0. We first construct an orthogonal set {w1,w2,w3} and will later reduce these to unit vectors {e1,e2,e3}, which will be the orthonormal basis. Take w1 = {2,0,0}. Let u2 = {3,4,0} - s{2,0,0} = {3-2s, 4, 0} To find s, we use the fact that we want u2.w1 to be 0. So: u2.w1 = 2(3-2s) + 0 + 0 = 0 if s = 3/2 Thus: u2 = {0,4,0} We take w2 = k * u2. We need to choose k so that elements of w2 are integers. This is satisfied with k = 1. Then: w2 = {0,4,0} Now let: u3 = {5,6,7} - s * w1 - t * w2 u3 = {5,6,7} - s{2,0,0} - t{0,4,0} u3 = {5-2s, 6-4t, 7} Again we use the fact that we want u3.w1 to be 0 and u3.w2 to be 0 to find s and t: u3.w1 = 2(5-2s) + 0 + 0 = 0 if s = 5/2 and u3.w2 = 0 + 4(6-4t) + 0 = 0 if t = 3/2 So u3 = {5,6,7} - (5/2){2,0,0} - (3/2){0,4,0} = {0,0,7} Taking w3 = u3 (because u3 consists of integers), we get: w3 = {0,0,7} So our orthogonal basis {w1,w2,w3} is given by: w1 = {2,0,0} w2 = {0,4,0} w3 = {0,0,7} And the orthonormal basis is: e1 = {1,0,0} e2 = {0,1,0} e3 = {0,0,1} - Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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