Matrix-vector Product - Advanced Engineering Math - Lecture Slides, Slides of Engineering Mathematics

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Matrix-vector product &

Linear Independence

Last lecture: Row Echelon Form

• A leading entry in a row means the first nonzero

entry from the left.

• A rectangular matrix is in row echelon form if

– All nonzero rows are above any all-zero row.

– All entries below a leading entry are zeros.

– In any pair of adjacent nonzero row, say row i and row

i+1, the leading entry in row i is to the left of the

leading entry of row i+1.

Imagine the red line as a stair

• All entries below the stair are zero
• All corner points are non-zero
• The width of the steps may vary
• We can only go down one row in one step

Rerview: Row and Column Vectors

• column vector

• row vector

• n -dimensional

column vector n

A list of n numbers written vertically

Convention: a vector is by default a column vector in this course

Convention: The components in a vector are sometime called “scalar”.

Illustration

x

y

Notation from Physics

x

y

has the same meaning as z

Interpretation of vector (I)

y

location

or

Interpretation of vector (3)

y

Any arrow in the same direction with the same length

or

Mathematical Notation

• The set of all 2-D vectors with real numbers as

components is denoted by

• The set of all 3-D vectors with real numbers as

components is denoted by

• The set of all n-D vectors with

real numbers as components:

Equality for vectors...

• … is just equality in each component

• Examples

Vector addition …

• … is just component-wise addition

But has no meaning.

Dot product

• A.k.a. scalar product, or inner product.

• For 2-D vector,

• It measures the “angle” between two vectors.

– The dot product of two vectors is zero if the two

vectors are perpendicular

Dot product in general

• For n-dimensional vectors in general, we

define the dot product as

• Example

Two n-dim vectors are said to be perpendicular, or orthogonal, if their dot product is equal to 0.

Matrix-vector multiplication

• Given an mn matrix A , and an n-dimensional

vector x , the product of A and x

is an m-dimensional vector defined as

For double subscripts, the first subscript is the row index and the second is the column index

Just compute dot products m times

Dot product of the first row in the matrix and the column vector

Dot product of the second row in the matrix and the column vector

Dot product of the last row in the matrix and the column vector