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By Kenneth L. Kuttler
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Example text
Ei ≡ 1 , .. . 0 where the 1 is in the ith position and there are zeros everywhere else. Thus T ei = (0, · · · , 0, 1, 0, · · · , 0) . Of course the ei for a particular value of i in Fn would be different than the ei for that same value of i in Fm for m ̸= n. One of them is longer than the other. However, which one is meant will be determined by the context in which they occur. These vectors have a significant property. 3 Let v ∈ Fn . Thus v is a list of numbers arranged vertically, v1 , · · · , vn .
However, which one is meant will be determined by the context in which they occur. These vectors have a significant property. 3 Let v ∈ Fn . Thus v is a list of numbers arranged vertically, v1 , · · · , vn . Then eTi v = vi . 20) makes perfect sense. It equals v1 . .. (0, · · · , 1, · · · 0) vi = vi .. . vn as claimed. 21). From the definition of matrix multiplication, and noting that (ej )k = δ kj ∑ A1j k A1k (ej )k . .. .. . ∑ T T T ei Aej = ei Aik (ej )k = ei Aij = Aij k ..
Then AT denotes the n × m matrix which is defined as follows. ( T) A ij = Aji The transpose of a matrix has the following important property. 17 Let A be an m × n matrix and let B be a n × p matrix. 17) is left as an exercise. 18 An n × n matrix A is said to be symmetric if A = AT . It is said to be skew symmetric if AT = −A. 50 CHAPTER 2. 19 Let 3 −3 . 7 1 3 0 2 −2 0 2 1 A= 1 5 3 −3 Then A is symmetric. 20 Let 0 A = −1 −3 Then A is skew symmetric. There is a special matrix called I and defined by Iij = δ ij where δ ij is the Kronecker symbol defined by { 1 if i = j δ ij = 0 if i ̸= j It is called the identity matrix because it is a multiplicative identity in the following sense.