N

If A is an mxn matrix, then the columns of A are linearly independent if and only if A has ……………… pivot columns

The columns of a matrix A are linearly independent if and only if Ax=0 has no free variables, meaning every variable is a basic variable, that is, if and only if every column of A is a pivot column

Why is the previous question true?

The statement is false. Take v1 and v2 to be multiples of one vector and take v3 to be not a multiple of that vector Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly dependent

The following statement is either true or false

If V1,V2,V3 are in R3 and V3 is not a linear combination of v1 v2, then {v1,v2,v3} is linearly independent

If V1,V2,V3 are in R3 and V3 is not a linear combination of v1 v2, then {v1,v2,v3} is linearly independent

trivial solution

An indexed set of vectors {v1…vp} is said to be linearly independent if the vector equation x1v1 + x2v2 +… =0

has only a ……

has only a ……

The weights must not all be zero such that c1v1+c2v2….=0

The set{v1….vp} is said to be linearly dependent if there exist weights c1…cp that are not all what?

The vector equation has only the trivial solution so the vectors are linearly independent

Determine if the vectors are linearly independent

[ 5 [7 [8

0 3 9

0] -6] -36]

[ 5 [7 [8

0 3 9

0] -6] -36]

homogenous

A system of linear equations is said to be what if it can be written an Ax=0

A zero solution

Trivial solution

– The homogenous equation Ax=0 has a nontrivial solution if and only if the equation has at least one free variable

Non-trivial solution

The augmented matrix of this set has a trivial solution so the set is linearly independent

Determine if the columns of the matrix form a linearly independent set

[0 -3 9

2 1 -7

-1 4 -4

1 -4 -2]

[0 -3 9

2 1 -7

-1 4 -4

1 -4 -2]

The columns of the matrix are linearly dependent because there are more vectors than entries

Determine if the columns of the matrix form a linearly independent set

[ 1 -3 4

-3 9 4]

[ 1 -3 4

-3 9 4]

The set is linearly independent because neither vector is a multiple of the other vector.

Determine by inspection whether the vectors are linearly independent

[-10 [2

-15 3

5] 1]

[-10 [2

-15 3

5] 1]

The values of h which makes the vectors linearly dependent are -96 because this will cause x3 to be a free variable

Find the values of h for which the vectors are linearly dependent

[1 [-3 [2

-3 10 1

-6] 6] h]

[1 [-3 [2

-3 10 1

-6] 6] h]

True. Alinear transformation is a function from Rn to Rm that assigns to each vector x in Rn a vector T(x) in Rm

True or false

A linear transformation is a special type of function

A linear transformation is a special type of function

false but it is true the other way

Every linear transformation is a matrix transformation

True. This equation correctly summarizes the properties necessary for a transformation to be linear

A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2

True; each image T(x) is of the form Ax. Thus the range is the set of all linear combinations of the columns of A

The range of the transformation x Ax is the set of all linear combinations of the columns of A

False; the question “is c in the range of T?” is the same as

does there exist an x whose image is c?

does there exist an x whose image is c?

If T: Rn to Rm is a linear transformation and if c is in Rm, then a uniqueness question is “Is c in the range of T”

True for a linear transformation T(0) is equal to 0

A linear transformation T: Rn to Rm always maps the origin of Rn to the origin of Rm

Given that the set { v1,v2,v3} is linearly dependent, there exist c1.c2,c3 not all zero, such that c1v1 + c2v2 +c3v3 = 0.

Let T: Rn to Rm be a linear transformation, and let {v1,v2,v3} be a linearly dependent set to Rn. Explain why the set {T(v1). T(v2), T(v3)} is linearly dependent

x = [ -25

-6

-1] The vector is unique because there are no free variables in the system of equations

-6

-1] The vector is unique because there are no free variables in the system of equations

If T is defined by T(x) =Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = [1 -4 3

0 1 2

2 -9 6

b=

[-4

-4

-2

0 1 2

2 -9 6

b=

[-4

-4

-2

9 rows and 6 columns

How many rows and columns must a matrix A have in order to define a mapping from R6 to R9 by the rule T(x) = Ax

Yes, b is in the range of the linear transformation because the system represented by the augmented matrix [A b] is consistent

Let b = [-1

1

0] and let A be the matrix [1 -2 5 -5

0 1 -3 5

2 -2 4 -4] Is b in the range of the linear transformation x to Ax

1

0] and let A be the matrix [1 -2 5 -5

0 1 -3 5

2 -2 4 -4] Is b in the range of the linear transformation x to Ax

T: let T:V to V be a linear transformation.

Then range of T is defined as, [T(x)] such x belongs to V

Then range of T is defined as, [T(x)] such x belongs to V

Range of Transformation

The correct image of [4

-3] is [15

4] The second image is

[3×1 -x2

7×1 +8×2]

-3] is [15

4] The second image is

[3×1 -x2

7×1 +8×2]

Let e1 = [1

0] let e2 = [0

1] let y1 =[3

7

y2 = [-1

8

and let T:R2 to R2 be a linear transformation that maps e1 into y1 and maps e2 into y2. Find the images of [4

-3] and [x1

x2]

0] let e2 = [0

1] let y1 =[3

7

y2 = [-1

8

and let T:R2 to R2 be a linear transformation that maps e1 into y1 and maps e2 into y2. Find the images of [4

-3] and [x1

x2]

T(e1) = y1

T(e2) = y2

T(e2) = y2

From the previous problem, the linear transformation T: R2 to R2 is defined as

cT(u) + dT(v)

As T is linear, for any scalars c and d T(cu +dv) =

[4 = 4[1 0] + -3[0 1] = 4e1 – 3e2

-3]

-3]

From the previous problem again, the vector [ 4 -3] can be written as

4y1 – 3y2

4[3 7] + -3[-1 8] = 15 4

4[3 7] + -3[-1 8] = 15 4

T([4 -3]) = T(4e1 -3e2)

shear transformation

Let A = [0 3

1 1] The transformation T:R2 to R2 defined by T(x) = Ax is called a

1 1] The transformation T:R2 to R2 defined by T(x) = Ax is called a

I) T(u+v) = T(u) + T(v) for all u, v in the domain of T

ii) T(cu) = cT(u) for all scalars c and all u in the domain of T

ii) T(cu) = cT(u) for all scalars c and all u in the domain of T

A transformation (or mapping) T is linear if

Transformation

A what T from Rn to Rm is a rule that assigns each vector x in Rn a vector T(x) in Rm

domain of T

The set Rn is called the

codomain of T

The set Rm is called the

jth

T(ej)

ej

A = [ T(e1)…… T(en)]

T(ej)

ej

A = [ T(e1)…… T(en)]

If T: Rn to Rm is a linear transformation, then there exists a unique matrix A such that the following equation is true

T(x) = Ax for all x in Rn

In fact, A is the mxn matrix whose…… column is the vector …. where……. is the jth column of the identity matrix Rn as shown in the following equation………………………….

T(x) = Ax for all x in Rn

In fact, A is the mxn matrix whose…… column is the vector …. where……. is the jth column of the identity matrix Rn as shown in the following equation………………………….

columns

The what of a matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution

dependent

A set of two vectors{v1, v2} is linearly ….. if at least one of the vectors is a multiple of the other.

independent

A set is linearly …….. if neither vectors are a multiple of the other

linearly dependent

If a set contains more vectors than there are entries in each vector, then the set is

linearly dependent

If a set S = {v1 …vp} in Rn contains the zero vector, then the set is

1.) solution

2.) linear combination

3.)Rm

4.) pivot position

2.) linear combination

3.)Rm

4.) pivot position

Let A be an mxn matrix. Then the following statements are logically equivalent.

-For each b in Rm, the equation Ax = b has a …………..

-Each b in Rm is a ………………….. of the columns of A

-The columns of A span ………….

A has a ………………. in each row

-For each b in Rm, the equation Ax = b has a …………..

-Each b in Rm is a ………………….. of the columns of A

-The columns of A span ………….

A has a ………………. in each row

linear combination

The equation Ax = b has a solution if and only if b is a ………… of the columns of A

c1v1 + c2v2 +…. cpvp

with c1…cp scalers

with c1…cp scalers

if v1….vp are in Rn then the set of all linear combinations of v1…. vp is denoted by Span {v1…vp) and is called the subset of Rn spanned by v1….vp. That is Span {v1… vp} is the collection of all vectors that can be written in the form

x1v1 + x2v2 ….. =b has a solution or the linear system’s augmented matrix has a solution [v1…..vp b]

Asking whether a vector b is in Span{v1…vp} amounts to asking whether the vector equation

pivot column

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a

x = su +tv

Parametric vector equation

1.) consistent

2.) homogenous equation

2.) homogenous equation

Suppose the equation Ax = b is ………….. for some given b and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p +vh where vh is any solution of the …………………… Ax = 0