SLIDE 1
Vectors
Defn. A matrix with one column is called a (column) vector. We use bold letters for vector variables, such as x and v. We sometimes write the column vector
- 3
5
- as
(3, 5).
vecMatONE: 2
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 - - 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
SLIDE 2
SLIDE 3
Vector Operations
Vector addition is performed by adding the cor- responding entries. Scalar multiplication is performed by scaling each entry. That is, u1 u2
- +
v1 v2
- =
u1 + v1 u2 + v2
- and
c u1 u2
- =
cu1 cu2
- For example
x 2 4
- + y
−1 7
- =
2x − y 4x + 7y
- vecMatONE: 3
SLIDE 4
Vectors and Points
Defn. We use Rd for the set of all d-entry vec- tors whose entries are real numbers. One can associate vector in Rd with the corre- sponding point. For example, R2 is the 2-dimensional plane. And vector addition can be illustrated with a parallelogram:
u v u + v
vecMatONE: 4
SLIDE 5
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
SLIDE 6
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
SLIDE 7
Matrix-Vector Multiplication
Defn. If A is an m × n matrix and x is in Rn, then the matrix-vector product Ax is the lin- ear combination of the columns of A specified by x. That is, if A = [a1, . . . , an] (meaning its columns are vectors a1, . . . , an), and x = (x1, . . . , xn) then Ax = x1a1 + x2a2 + . . . + xnxn
vecMatONE: 7
SLIDE 8
Example of Matrix-Vector Multiplication
For example, 2 −1 4 7 3 5
- = 3
2 4
- + 5
−1 7
- =
1 47
- vecMatONE: 8
SLIDE 9
Summary
A vector is a matrix with one column. We use bold letters for vector variables. Rd is all d- 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