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

Suppose it is given that m is a positive integer and that T: R^3 --> R^m is 1-1. Which cannot be m?

1

2

3

4

5

Since it is 1-1 we need each col to be LI

n = 3, m = ? so its a

mx3 matrix

since 3 cols we know

m>=3 and therefore

1,2, cannot be values

Suppose S:R^3 --> R^3 and T: R^3 --> R^3 are linear transformations satisfying S(T(v)) = v and T(S(v)) = v for all vectors in the span of R^3. Given that the standard matrix of T has a determinant of -2, what is the determinant of the standard matrix of S^2?

Since S = T^-1

We do det(T^-1) = 1/det(T)

1/-2 = S

S^2 = 1/4

Let T : R3 → R2 be the linear transformation given by T ((x y z)T ) = (2x + ay + 4z 3x − 9y + bz)T where a, b are scalars. Suppose it is given that T is not onto. Find the only possible value of b.

1. make the coeff matrix

2 a 4

3 -9 b

2. set R1*k to R2

3 = 2k

-9 = ak

b = 4k

3. solve for k with 3=2k

k = 3/2

4. solve for a, b

a = -6

b = 6

therefore

b = 6

A value is not invertible if

its determinant = 0

Let T:R^2→R^2 be a non-zero linear transformation such that

T^2(v) = 0 for all v in R, and suppose T([0,1]) = [1,2]. what is the sum of the entries of the standard matrix A of T?

e_1 = [1,0] and e_2 = [0,1]

we know Te_2 = [1,2] and is the 2nd col of A

we know Te_1 = [a,b] and is the first col of A

2. set A = a 1

b 2

3. use the fact that since T^2 = 0, A^2 = 0 and solve for a and b

| a 1 | | a 1 | = | 0 0 |

| b 2 | | b 2 | | 0 0 |

we get

a^2 + b, a + 2

ab + 2b, b + 4

a^2 + b =0

a + 2 = 0

ab + 2b = 0

b + 4 = 0

a =-2 b = -4

4. sub in the a's and the b's and get the sum

= -3

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

Suppose T : R3 → R2 is a linear transformation and

T(( 1, 2, 3)) = (-2, 4) and T((2, 0, -1)) = (1, -2)

'

Among the vectors below, which must be non-trivial solutions to T (x) = 0? Select all thatapply.

Not gonna list em, just need to know how to solve

1. gather what we know

v = [ 1, 2, 3] and v_2 = [2 0 -1]

T(v_1) = [-2, 4] and T(v_2) = [1, -2]

we know

T(x) = aT(v1) + aT(v2) = 0

2. write in numbers form

and solve for a and b

a[-2, 4] + b[1, -2] = 0

getting us

-2a + b = 0

4a = 2b = 0 ----> b = 2a

3. rewrite to solve

x = av1 + 2av2 = a(v1 + 2v2) = 0

then we can do

[1 2 3] + 2[2 0 -1] = [1+4, 2 + 0, 3 -2] = [5, 2, 1]

so it equals the column vector

5

2

1

and therefore

Null(T) = span{(5, 2, 1)}

so any scalar mult we would select

The linear transformation T : R2 → R2 does the following, in order:

1. reflects a vector vabout the line y = x

2. projects the resulting vector orthogonally onto u = ( 1 2)^T

3. then reflects the projected vector about y = x again.

Find the value of c such that T is orthogonal projection onto (1 c )^T

1. get what we know

u = [1 , 2]

1. reflect about y = x

R = [ 0 1

1 0 ]

3. orthog project onto u

P = (u u^T)/ (u^t u) = 1/ (1^2 + 2^2) * [1, 2]

= 1/5 * [ 1 2 ]

[ 2 4 ]

4. apply the full transformation

T = R P_u R

We know R and P

=

(1/5) [ 4 2 ]

[ 2 1 ]

tbh im giving up

do the same thing for c

gets u c = 0.5 if u do the proj

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

Example:

Determine the null space of

A = 0 1 2 3

0 2 4 6

0 3 6 9

0 -1 -2 -3

1. row reduce to get the # of pivots.

the number of pivots is the rank

rank(A) = 1

2. use Rank-Nullity Theorem

rank + nullity = n

1 = nullity = 4

null space = 3

One can show that V = {(x,y,z) is in domain R^3 | x^2 + 2y + 2a = 0} is a subspace of R^3

Let V be the standard matrix of the linear transformation T:R63 --> R^3 that orthogonally projects vectors onto the subspace V . Determine the rank of A

We know that

1. dim(v) = n-k

k is 1 as there is only 1 equation

dim(v) = 3-1

dims(v) = 2

2. we know that

Rank(A) = dim(V)

therefore rank(A) = 2

or exercises 1 and 2, use the following matrix A and its given reduced row echelon form

A = 1 -2 0 -1 3,

0 0 1 2 -2

0 0 0 0 0

The column space of a matrix is the span of its columns. Which of the following is a basisfor the column space of A?

1. count the # of pivots

there are 2

2. we have 2 pivots so set those 2 lines to 0

solve for the x1, x2, x3, etc

find the free vars and set them as x2() + x3() + x5() = x with each instance that its used on a non-free var, and 1 at its own instance

he null space of a matrix is the set of solutions to the homogeneous system with the matrixas its coefficient matrix. Which of the following is a basis for the null space of A

A = 1 -2 0 -1 3,

0 0 1 2 -2

0 0 0 0 0

1. count the mxn dims

its 3x5 meaning the basis u choose must have 3 LI vectors each w five rows

2. count the pivot cols (2)

take the coeffs, assign x1, x2, etc. set = 0

solve for

Let v1, v2, vk be vectors in R^n. The subspace V = Span(v1, v2, vk) of R^n is said to be generasted by the v's v1 v2 vk

What is the dimension of the subspace of R^3 generated by the following vectors?

( 1, 2, 3), ( 0, 2, -1), (2, 0, 8), (1, 0, 4), (4, 4, 14)?

1. put them all in a big coeff matrix

2. row reduce to find the rank and check for any LD cols

(1 0 4) is LD of (2 0 8) so we can drop it

3. write v5 as a linear combo of the others

v5 = av1 + bv2 + cv4

4=a(2)+b(2)+c(0)=2a+2b

14=a(3)+b(−1)+c(4)=3a−b+4c

solve the letters

re plug in

4. set = 0

a(0,2,1) + b(1 0 4) = (0 0 0)

so ld vectors is 2

=2

Let V be the subspace of R3 with (ordered) basis C = {(3 − 2 7)T , (1 − 2 2)T }. Find the second component of the coordinates of the vector (1 2 3)T ∈ V relative to the basis C

v1 = (3, -2, 7)

v2 = (1, -2, 2)

w = av1 + bv2 = a(3, -2, 7) + b(1, -2, 2)

3a + b,

-2a + 2b

7a + 2b

solve for b = -2

then solve for a = 1

find the second component

-2(-1) + 2(-2)

=-2

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

Suppose A is a matrix with real entries, with a rank of 5 and that Null(A) = span( ⃗v 1, ⃗v 2)for some vectors ⃗v 1, ⃗v 2, with the same number of components and satisfying that neither isa scalar multiples of the other. Determine the number of columns A has

1. use rank nullity theorem

rank(A) + dims(Null(A))

we know dims(Null(A)) is 2 since only spans 2 v's

5 + 2 = 7

=7

A square matrix P that satisfying P 2 = P is called a "projection" matrix. What is themaximum possible rank of a 4 × 4 non-identity projection matrix P ? Justify your answerby producing an example of such a 4 × 4 matrix

P can be 4, but since we need a lower dimensinoal subspace =3

Suppose U and V are distinct subspaces of R7 that are both of dimension 4. WhichCANNOT be the dimension of U + V ? Select all that apply.Hint: the dimension of U + V is the rank of a 7 × 8 matrix where the first four columnsform a basis of U and the last four columns form a basis of V .

4 + 4 = 8 minus instance of both so 7

0-7 yes

anything else no

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).