W6 W7 Linear Algebra OSU

Linear transformation

T:R^n ---> R^m

maps vectors from the R^n domain

to the R^m codomain

it has range of T: the set of all images T(x)

What makes a linear transformation linear?

1. T(u+v) = T(u) + T(v)

2. T(cu) = cT(u)

Matrix Transformation

T(x) = Ax

A is an m x n matrix

It maps vectors in R^n to vectors in R^m by

multiplying them with matrix A

What does one-to-one mean for linear transformations?

Each output corresponds to at most one input

Ex: Tx_1 = Tx_2 ---> x1 = x2

and T(X) = 0

2. all the columns are linearly independent of each other

Think like this matrix

1 0 0

0 1 0

0 0 1 only 1 possible input for each row

A linear transformation is onto if?

1. Each vector in R^m is the image of some vector R^n

2. The matrix A has a pivot in every row

Intertible linear transformations:

T:R^n --> R^n is invertibe if?

There exists S such that

1. T(S(x)) = x

2. S(T(x)) = x

3. T is invertible ( T is 1-1 and onto)

4. the matrix A is invertible

5. the inverse transformation T^-1 has matrix A^-1

6. T = S^-1 and T^-1 = S

A value is not invertible if

its determinant = 0

T is one-to-one if and only if

The matrix A has a pivot position in every column

T is one-to-one but not onto if .

The matrix A has n pivots and m > n

What is a subspace?

What makes a subspace?

A subspace is a subset of v in the domain of R that

have the following properties:

1. Zero vector is in V

2. it is closed under addition, meaning

u + v and in the domain of V for all u and v vals

3. it is closed under scalar multiplication

c * v in in the domain of V for all v in V and c in V

Subspace properties

1. the null space of any matrix A is a subspace of R^n

2. Col(A) and Row(A) are subspaces of R^m and R^n repsectively

3. Pivot columns of A (not RREF(A)) form a basis for Col(A)

What is a basis of a subspace V?

a set of vectors that

1. Span the space V

2. are linearly independent,

each col has to be LI from the others

coordinates relative to a basis

x = c1b1 + ... + ckbk

c1, c2, cn, etc are called the coords of x relative to basis B

|x|_B = [ c1, c2, ..., ck ]

example:

B = {[ 1, 0] , [ 0,1]}

x = [4, 3]

x = 4[1, 0] + 3[0,1] ---> |x|_B = [4,3]

What is the dimension of a subspace?

The number of vectors in any basis of the subspace. It tells you how many "directions" the space has

What is the null space of a matrix A?

All vectors x such that Ax = 0. it is a subspace of the input space

what is the rank of a matrix A?

The dimension of the column space of A, i.e., the number of linearly independent columns.

What is the Rank-Nullity Theorem?

For a = m x n

rank(A) + nullity(A) = n

What can be said about non-pivot columns of A?

Each non-pivot column is a linear combination of pivot columns

State the Basis Theorem for a k-dimensional subspace V in the domain of R

Any set of k linearly independent vectors in V is a basis for V. Also, any set of k vectors that spans V is a basis.

What does the Invertible Matrix Theorem say about the null space if A is invertible?

Null(A) = 0

What is a coordinate vector |X|_B relative to a basis B= {u1..., uk}?

It's the vector of scalars (x1, x2, xk) such that x = sum of xiui

How do you convert coordinates from basis B to basis C?

Use the change of basis matrix A with columns |u1|c, |u2|c, ... |uk|c then

|x|_c = A|x|_B

What is the relation between the change of basis matrices from B→C and C→B

They are inverses of each other:

AB = BA = I_k

Are change of basis matrices invertible?

Yes. The inverse of the change of basis matrix from B to C is the matrix from C to B

What is the standard basis S for R^n?

S = {e1, e2,.... en} the columns of the identity matrix I_n

What is the change of basis matrix from basis B to standard basis S

The matrix whose columns are the vectors in B (in order).

What is the change of basis matrix from standard basis S to basis C?

It is the inverse of the matrix whose columns are the vectors in C

Trace of a matrix

The trace of a square matrix is the sum of the entries on its main diagonal (from top left to bottom right).

T or F

A change of basis matrix is an invertible square matrix

True

T or F:

There exists a singular change of basis matrix

False

T or F:

Any invertible square matrix is a change of basis matrix

True

T or F:

The inverse of a change of basis matrix exists and is a change of basis matrix.

True

T or F:

There exist non-square change of basis matrices

False

What is a standard matrix?

The matrix A such that T(x) = Ax for any vector x in the domain R^n

To find it, apply T to each standard basis vector. use the resulting vectors as columns for the matrix

What is a standard basis vector?

the set of n vectors

e1, e2, ..., en

example:

5

-2

4

is 5e_1 -2e_2 + 4e_3

What is a change of basis matrix?

a matrix used to convert the coordinates of vectors from one basis to another

Columns = vectors of the old basis, expressed in the new basis

Example (dont necessarily have to know)

ex: B = { b1, b2, bn} S = {e1, e2, en}

Then the change of basis matrix from basis B to standard basis S is the matrix:

P = [b1 b2 bn]

to get

|x|_c = Q^-1 * P|x|_B