Vectors and Matrices Vectors Defn. A matrix with one column is - PowerPoint PPT Presentation

Vectors and Matrices

Vectors Defn. A matrix with one column is called a (column) vector . We use bold letters for vector variables, such as x and v . � � 3 We sometimes write the column vector as 5 (3 , 5) . vecMatONE: 2

Vector Operations Vector addition is performed by adding the cor- responding entries. Scalar multiplication is performed by scaling each entry. That is, � u 1 � � v 1 � � u 1 + v 1 � � u 1 � � cu 1 � + = and = c u 2 + v 2 u 2 v 2 u 2 cu 2 For example � 2 x − y � 2 � � − 1 � � + y = x 4 7 4 x + 7 y vecMatONE: 3

Vectors and Points We use R d for the set of all d -entry vec- Defn. tors whose entries are real numbers. One can associate vector in R d with the corre- sponding point. For example, R 2 is the 2 -dimensional plane. And vector addition can be illustrated with a parallelogram: u + v v u vecMatONE: 4

Linear Combinations Defn. A linear combination of vectors is formed by summing some multiple of each vec- tor. The multipliers are called the weights . vecMatONE: 5

Spans Defn. The span of a collection of vectors is the set of all possible linear combinations. If S is a set, we will denote its span by Span S . For example, the span of a single (nonzero) vec- tor is a line. The span of two vectors is (usually) a plane. vecMatONE: 6

Matrix-Vector Multiplication If A is an m × n matrix and x is in R n , Defn. then the matrix-vector product A x is the lin- ear combination of the columns of A specified by x . That is, if A = [ a 1 , . . . , a n ] (meaning its columns are vectors a 1 , . . . , a n ), and x = ( x 1 , . . . , x n ) then A x = x 1 a 1 + x 2 a 2 + . . . + x n x n vecMatONE: 7

Example of Matrix-Vector Multiplication For example, � 1 � 2 − 1 � � 3 � � 2 � � − 1 � � = 3 + 5 = 4 7 5 4 7 47 vecMatONE: 8

Summary A vector is a matrix with one column. We use R d is all d - bold letters for vector variables. entry vectors with real entries. Vector addition adds corresponding entries; scalar multiplica- tion scales each entry. A linear combination of vectors is any sum of some multiple of each vector. Their span is the set of all possible linear combinations. The prod- uct of matrix A with vector x is the linear com- bination of columns of A given by x . vecMatONE: 9