Math 2270 Spring 2004 Basic Notation and Symbols Standard Notations
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Math 2270 Spring 2004 Basic Notation and Symbols Standard Notations Scalars are generally represented by lower case letters with no additional notation: a, b, c, d · · ·. Vectors are simply lists of numbers written either in a column or row. They are generally represented by lower case letters with an arrow on top: ~u, ~v , w, ~ ~x, ~y , ~z · · ·. The individual components of a vector are generally represented by lower case letters with numerical subscripts indicating which position they hold in the vector: vector ~x has components x1 , x2 , x3 · · ·. Matrices are arrays of numbers written in rows and columns simultaneously. Notice that a vector is a special type of matrix; one with only one row or only one column. Matrices are generally represented by capital letters: A, B, C, D · · ·. Shorthand ∀ = “for all” or “for every” ∃ = “there exists” if f = “if and only if” ∈ = “an element of” or “a member of the group of” s.t. = “such that” ℜ = the set of all the real numbers or scalars 1 4.24 −5 9 7 π ℜ2 = the set of all vectors with two real number entries " 1 5 # " 2.56 π # " −51 93.7 # ℜn = the set of all vectors with n real number entries x1 x2 x3 ··· xn 1 1 2.56 π ··· 10 " 1/3 7/4 # Math 2270 Spring 2004 Basic Notation and Symbols Matrix Basics We can look at a matrix in a variety of different ways. First, we can think of it as an array of real numbers. Here is an example of a 4x3 matrix with 12 entries. A= a11 a21 a31 a41 a12 a22 a32 a42 a13 a23 a33 a43 = 1 2 3 4 5 6 7 8 9 10 11 12 Second, we can think of a matrix as a bunch of column vectors lined up next to one another. We can write matrix A (from above) as three column vectors: ~v1 = 1 2 3 4 ~v2 = A = h 5 6 7 8 ~v3 = v~1 v~2 v~3 9 10 11 12 i The vectors ~v1 , ~v2 , ~v3 ∈ ℜ4 because they each have four entries. Third, we can think of a matrix as a bunch of row vectors stacked on top of one another. We can write matrix A as a stack of four row vectors: w ~1 = h 1 5 9 i w ~2 = h 2 6 10 i w ~3 = h 3 7 11 i w ~4 = h 4 8 12 i A = w ~1 w ~2 w ~3 w ~4 The vectors w ~ 1, w ~ 2, w ~ 3, w ~ 4 ∈ ℜ3 because they each have three entries. 2 Math 2270 Spring 2004 Basic Notation and Symbols Matrix-Vector Multiplication Now that we’ve looked at the matrix on its own, let’s look at what happens when we multiply a matrix by a vector. We can think of matrix-vector multiplication in three different ways, corresponding to the three different matrix representations mentioned previously. First, we can multiply the matrix and vector directly. A~x = 1 2 3 4 5 6 7 8 9 10 11 12 x1 x2 = x3 1x1 + 5x2 + 9x3 2x1 + 6x2 + 10x3 3x1 + 7x2 + 11x3 4x1 + 8x2 + 12x3 Second, we can rewrite the product in vector notation. We can think of the result as a linear combination of the column vectors of the matrix A, where the scalar coefficients are the entries of the vector ~x. A~x = x1~v1 + x2~v2 + x3~v3 = x1 1 2 3 4 + x2 5 6 7 8 + x3 9 10 11 12 = 1x1 + 5x2 + 9x3 2x1 + 6x2 + 10x3 3x1 + 7x2 + 11x3 4x1 + 8x2 + 12x3 Third, we can think of the product as a series of dot product calculations. A~x = w ~ 1 · ~x w ~ 2 · ~x w ~ 3 · ~x w ~ 4 · ~x = [1 5 9] · [x1 x2 x3 ] [2 6 10] · [x1 x2 x3 ] [3 7 11] · [x1 x2 x3 ] [4 8 12] · [x1 x2 x3 ] 3 = 1x1 + 5x2 + 9x3 2x1 + 6x2 + 10x3 3x1 + 7x2 + 11x3 4x1 + 8x2 + 12x3 Math 2270 Spring 2004 Basic Notation and Symbols Special Matrices and Vectors The identity matrix, In , is an nxn matrix with ones on the diagonal and zeros everywhere else. I1 = 1 I2 = " 1 0 0 1 1 0 0 I3 = 0 1 0 0 0 1 # I5 = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 The columns of the identity matrix are assigned their own vector names. Column VectorName one two ~e1 ~e2 ℜ1 ℜ2 [1] " − " ℜ3 1 0 # 0 1 # three ~e3 − − four ~e4 − − 1 0 0 0 1 0 0 0 1 − ℜ4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Notice that the same notation for a column (~ei ) is used no matter what space we are talking about (ℜ1 , ℜ2 , ℜn ). The appropriate length must be inferred from the context. 4
