Linear algebra and differential equations (Math 54): Lecture 3 - PowerPoint PPT Presentation

An example with numbers   w     3 5 4 3 2 x   2 1 1 1  = 6       y  1 1 − 1 1 − 3 z Expanding the product on the left, we find     3 w + 5 x + 4 y + 3 z 2  = 2 w + x + y + z 6    w + x − y + z − 3

An example with numbers   w     3 5 4 3 2 x   2 1 1 1  = 6       y  1 1 − 1 1 − 3 z Expanding the product on the left, we find     3 w + 5 x + 4 y + 3 z 2  = 2 w + x + y + z 6    w + x − y + z − 3 or in other words 3 w + 5 x + 4 y + 3 z = 2 2 w + x + y + z = 6 w + x − y + z = − 3

A x = b If we write       · · · a 11 a 12 a 1 c x 1 b 1 a 21 a 22 · · · a 2 c x 2 b 2       A = x = b =  . . .   .   .  ... . . . . .  ,  ,       . . . . .     · · · a r 1 a r 2 a rc x c b r

A x = b If we write       · · · a 11 a 12 a 1 c x 1 b 1 a 21 a 22 · · · a 2 c x 2 b 2       A = x = b =  . . .   .   .  ... . . . . .  ,  ,       . . . . .     · · · a r 1 a r 2 a rc x c b r Then the equation       · · · a 11 a 12 a 1 c x 1 b 1 · · · a 21 a 22 a 2 c x 2 b 2        =  . . .   .   .  ... . . . . .       . . . . .      · · · a r 1 a r 2 a rc x c b r

A x = b If we write       · · · a 11 a 12 a 1 c x 1 b 1 a 21 a 22 · · · a 2 c x 2 b 2       A = x = b =  . . .   .   .  ... . . . . .  ,  ,       . . . . .     · · · a r 1 a r 2 a rc x c b r Then the equation       · · · a 11 a 12 a 1 c x 1 b 1 · · · a 21 a 22 a 2 c x 2 b 2        =  . . .   .   .  ... . . . . .       . . . . .      · · · a r 1 a r 2 a rc x c b r becomes simply A x = b .

A x = b If we write       · · · a 11 a 12 a 1 c x 1 b 1 a 21 a 22 · · · a 2 c x 2 b 2       A = x = b =  . . .   .   .  ... . . . . .  ,  ,       . . . . .     · · · a r 1 a r 2 a rc x c b r Then the equation       · · · a 11 a 12 a 1 c x 1 b 1 · · · a 21 a 22 a 2 c x 2 b 2        =  . . .   .   .  ... . . . . .       . . . . .      · · · a r 1 a r 2 a rc x c b r becomes simply A x = b . Here A is called the matrix of coefficients.

The matrix-vector product The product A x can also be written as         · · · a 11 a 12 a 1 c x 1 a 11 a 1 c a 21 a 22 · · · a 2 c x 2 a 21 a 2 c          = x 1  + · · · + x c  . . .   .   .   .  ... . . . . . .         . . . . . .       · · · a r 1 a r 2 a rc x c a r 1 a rc

The matrix-vector product The product A x can also be written as         · · · a 11 a 12 a 1 c x 1 a 11 a 1 c a 21 a 22 · · · a 2 c x 2 a 21 a 2 c          = x 1  + · · · + x c  . . .   .   .   .  ... . . . . . .         . . . . . .       · · · a r 1 a r 2 a rc x c a r 1 a rc That is, A x is a linear combination

The matrix-vector product The product A x can also be written as         · · · a 11 a 12 a 1 c x 1 a 11 a 1 c a 21 a 22 · · · a 2 c x 2 a 21 a 2 c          = x 1  + · · · + x c  . . .   .   .   .  ... . . . . . .         . . . . . .       · · · a r 1 a r 2 a rc x c a r 1 a rc That is, A x is a linear combination of the columns of A ,

The matrix-vector product The product A x can also be written as         · · · a 11 a 12 a 1 c x 1 a 11 a 1 c a 21 a 22 · · · a 2 c x 2 a 21 a 2 c          = x 1  + · · · + x c  . . .   .   .   .  ... . . . . . .         . . . . . .       · · · a r 1 a r 2 a rc x c a r 1 a rc That is, A x is a linear combination of the columns of A , with coefficients given by the entries of the vector x . The equation A x = b

The matrix-vector product The product A x can also be written as         · · · a 11 a 12 a 1 c x 1 a 11 a 1 c a 21 a 22 · · · a 2 c x 2 a 21 a 2 c          = x 1  + · · · + x c  . . .   .   .   .  ... . . . . . .         . . . . . .       · · · a r 1 a r 2 a rc x c a r 1 a rc That is, A x is a linear combination of the columns of A , with coefficients given by the entries of the vector x . The equation A x = b asserts that the vector b

The matrix-vector product The product A x can also be written as         · · · a 11 a 12 a 1 c x 1 a 11 a 1 c a 21 a 22 · · · a 2 c x 2 a 21 a 2 c          = x 1  + · · · + x c  . . .   .   .   .  ... . . . . . .         . . . . . .       · · · a r 1 a r 2 a rc x c a r 1 a rc That is, A x is a linear combination of the columns of A , with coefficients given by the entries of the vector x . The equation A x = b asserts that the vector b is equal to this linear combination of the columns of A .

An example with numbers   w     3 5 4 3 2 x   2 1 1 1  = 6       y  1 1 − 1 1 − 3 z

An example with numbers   w     3 5 4 3 2 x   2 1 1 1  = 6       y  1 1 − 1 1 − 3 z The product on the left means a linear combination of the columns of the matrix, weighted by the entries of the vector.  3   5   4   3   2   + x  + y  + z  = w 2 1 1 1 6       1 1 − 1 1 − 3

An example with numbers   w     3 5 4 3 2 x   2 1 1 1  = 6       y  1 1 − 1 1 − 3 z The product on the left means a linear combination of the columns of the matrix, weighted by the entries of the vector.  3   5   4   3   2   + x  + y  + z  = w 2 1 1 1 6       1 1 − 1 1 − 3 or in other words 3 w + 5 x + 4 y + 3 z = 2 2 w + x + y + z = 6 w + x − y + z = − 3

A function from R 4 to R 3   w x     y   z

A function from R 4 to R 3     w w   3 5 4 3 x x      �→ 2 1 1 1       y y    1 1 − 1 1 z z

A function from R 4 to R 3     w w     3 5 4 3 3 w + 5 x + 4 y + 3 z x x      �→ 2 1 1 1  = 2 w + x + y + z         y y   1 1 − 1 1 w + x − y + z z z

A function from R 4 to R 3     w w     3 5 4 3 3 w + 5 x + 4 y + 3 z x x      �→ 2 1 1 1  = 2 w + x + y + z         y y   1 1 − 1 1 w + x − y + z z z This map brought to you by the matrix   3 5 4 3 2 1 1 1   1 1 − 1 1

Functions from matrices If A is a matrix with

Functions from matrices If A is a matrix with r rows and c columns,

Functions from matrices If A is a matrix with r rows and c columns, and x ∈ R c is any vector,

Functions from matrices If A is a matrix with r rows and c columns, and x ∈ R c is any vector, then A x ∈ R r .

Functions from matrices If A is a matrix with r rows and c columns, and x ∈ R c is any vector, then A x ∈ R r . So the matrix A defines a function (also called A ):

Functions from matrices If A is a matrix with r rows and c columns, and x ∈ R c is any vector, then A x ∈ R r . So the matrix A defines a function (also called A ): A : R c R r → x �→ A x

Functions from matrices If A is a matrix with r rows and c columns, and x ∈ R c is any vector, then A x ∈ R r . So the matrix A defines a function (also called A ): A : R c R r → x �→ A x The equation A x = b can be read as:

Functions from matrices If A is a matrix with r rows and c columns, and x ∈ R c is any vector, then A x ∈ R r . So the matrix A defines a function (also called A ): A : R c R r → x �→ A x The equation A x = b can be read as: which vectors x have the property that their image under the function A is the vector b ?

Functions from matrices This is similar to the way that a number a determines a function “multiplication by a ”

Functions from matrices This is similar to the way that a number a determines a function “multiplication by a ” a : R → R �→ x ax

Functions from matrices This is similar to the way that a number a determines a function “multiplication by a ” a : R → R �→ x ax Indeed, this is the case n = m = 1.

Linearity The matrix-vector product has the following properties: A ( x + y ) = A x + A y A ( c x ) = c ( A x )

Linearity The matrix-vector product has the following properties: A ( x + y ) = A x + A y A ( c x ) = c ( A x ) Functions with these two properties are said to be linear.

Linearity The matrix-vector product has the following properties: A ( x + y ) = A x + A y A ( c x ) = c ( A x ) Functions with these two properties are said to be linear. In fact, every linear function from R c to R r

Linearity The matrix-vector product has the following properties: A ( x + y ) = A x + A y A ( c x ) = c ( A x ) Functions with these two properties are said to be linear. In fact, every linear function from R c to R r is multiplication by some matrix.

Linearity The matrix-vector product has the following properties: A ( x + y ) = A x + A y A ( c x ) = c ( A x ) Functions with these two properties are said to be linear. In fact, every linear function from R c to R r is multiplication by some matrix. We will see why this is next time.

Thinking about linear equations

Thinking about linear equations We have seen that linear equations can be interpreted as

Thinking about linear equations We have seen that linear equations can be interpreted as ◮ Asking for the intersection locus of lines or planes

Thinking about linear equations We have seen that linear equations can be interpreted as ◮ Asking for the intersection locus of lines or planes ◮ Asking how to write one vector as a linear combination of given others

Thinking about linear equations We have seen that linear equations can be interpreted as ◮ Asking for the intersection locus of lines or planes ◮ Asking how to write one vector as a linear combination of given others ◮ Asking for vectors mapping to a given one under the linear function associated to the coefficient matrix

Thinking about linear equations We have seen that linear equations can be interpreted as ◮ Asking for the intersection locus of lines or planes ◮ Asking how to write one vector as a linear combination of given others ◮ Asking for vectors mapping to a given one under the linear function associated to the coefficient matrix We know how to compute the solutions

Thinking about linear equations We have seen that linear equations can be interpreted as ◮ Asking for the intersection locus of lines or planes ◮ Asking how to write one vector as a linear combination of given others ◮ Asking for vectors mapping to a given one under the linear function associated to the coefficient matrix We know how to compute the solutions (row reduction).

Thinking about linear equations We have seen that linear equations can be interpreted as ◮ Asking for the intersection locus of lines or planes ◮ Asking how to write one vector as a linear combination of given others ◮ Asking for vectors mapping to a given one under the linear function associated to the coefficient matrix We know how to compute the solutions (row reduction). Now we’ll discuss qualitative properties of the solution set.

Homogenous and inhomogenous equations

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b ,

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 )

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0 That is, the difference between two solutions to the inhomogenous equation A x = b

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0 That is, the difference between two solutions to the inhomogenous equation A x = b is a solution to the homogenous equation A x = 0.

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0 That is, the difference between two solutions to the inhomogenous equation A x = b is a solution to the homogenous equation A x = 0. Differently said,

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0 That is, the difference between two solutions to the inhomogenous equation A x = b is a solution to the homogenous equation A x = 0. Differently said, given one solution x = x 0 to the inhomogenous equation A x = b ,

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0 That is, the difference between two solutions to the inhomogenous equation A x = b is a solution to the homogenous equation A x = 0. Differently said, given one solution x = x 0 to the inhomogenous equation A x = b , all other solutions are given by the sum of x 0

Homogenous and inhomogenous equations If A x 1 = b and A x 2 = b , then A ( x 1 − x 2 ) = A x 1 − A x 2 = b − b = 0 That is, the difference between two solutions to the inhomogenous equation A x = b is a solution to the homogenous equation A x = 0. Differently said, given one solution x = x 0 to the inhomogenous equation A x = b , all other solutions are given by the sum of x 0 and a solution to the homogenous equation A x = 0.

Try it yourself Find all solutions to the homogenous equation x + y = 0 and to the inhomogenous equation x + y = 1. Graph your answers. How do they compare?

Homogenous equations and linearity If A x 1 = 0 and A x 2 = 0,

Homogenous equations and linearity If A x 1 = 0 and A x 2 = 0, then for any scalars c 1 , c 2 ,

Homogenous equations and linearity If A x 1 = 0 and A x 2 = 0, then for any scalars c 1 , c 2 , A ( c 1 x 1 + c 2 x 2 )

Homogenous equations and linearity If A x 1 = 0 and A x 2 = 0, then for any scalars c 1 , c 2 , A ( c 1 x 1 + c 2 x 2 ) = c 1 A x 1 + c 2 A x 2

Homogenous equations and linearity If A x 1 = 0 and A x 2 = 0, then for any scalars c 1 , c 2 , A ( c 1 x 1 + c 2 x 2 ) = c 1 A x 1 + c 2 A x 2 = 0

Homogenous equations and linearity If A x 1 = 0 and A x 2 = 0, then for any scalars c 1 , c 2 , A ( c 1 x 1 + c 2 x 2 ) = c 1 A x 1 + c 2 A x 2 = 0 In other words,