Linear 2

Find a basis for the column space of 3 3 -2 0 0 4 -4 2 -3 1 -2 2

3 3 0 4 -3 1

two special subspaces

A mxn matrix 1. column A (columns pace of A): span of the columns of A (subspace of R^m) 2. null A (null space of A): solution set to Ax=0 (solution sets are subspaces of R^n)

Let T: R2→R2 be the function given by T x |x| y = |y| T(u+v) = T(u) + T(v) for all vectors u and v.

F

Let V be the subset of R3 consisting of the vectors a b c with abc=0. V is closed under vector addition, meaning that if u and v are in V then u+v is in V.

F

The null space of an m×n matrix is a subspace of Rm.

F

basis for null A

(solution set to Ax=0) use paramedic vector form of a solutions et to Ax=0

Which sets of vectors are linearly independent? 2 3 5 -8 9 -6 0 0 0 -1 2 3 1 9 -8 1 4 5 0 -2 -6 7 0 4 0 -8 14 -7 12 -6 -3 -4 7 -2 8 6

-1 2 3 1 9 -8 1 4 5 0 -2 -6 7

Let V be the subspace spanned by {w1,w2,w3}. Find a linearly independent set of vectors {v1,v2} that spans V, so that neither v1 nor v2 is a scalar multiple of any of w1,w2,w3. -3 1 0 5 -4 -7 -4 -2 -10

-4 -2 9 1 -2 -6

Let T:R3→R2 be the linear transformation that first projects points onto the xy-plane (forgetting the z coordinate entirely) and then reflects around the line y=−x. Find the standard matrix A for T.

0 -1 0 -1 0 0

Let A= −1 −1 1 1 −1 1 −1 −1 1 1 1 −1 1 −1 −1 Find a non-zero vector x in the null space of A.

1 1 1 1 0

Find a 3×3 matrix A with the following property: if T(x)=Ax, then the range of T is the xz-plane.

1 0 0 0 0 0 0 0 1

Find examples of a 4×4 matrix A such that T is not onto.

1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

Find examples of a 4×4 matrix A such that T is onto.

1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0

Find examples of a 4×4 matrix A such that T is not 1-1.

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0

Find examples of a 4×4 matrix A such that T is 1-1.

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

Find four vectors v1,v2,v3,v4 in R4 such that no two are collinear (in particular, none of the vectors is zero) the set {v1,v2,v3,v4} is linearly dependent, and v4 is not in

1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0

Match the following concepts with the correct definitions: 1. T is an onto transformation from R3 to R3 2. T is a one-to-one transformation from R3 to R3 3. T is a transformation from R3 to R3 A. For every y in R3, there is a x in R3 such that T(x)=y. B. For every y in R3, there is at most one x in R3 such that T(x)=y. C. For every y in R3, there is a unique x in R3 such that T(x)=y. D. For every x in R3, there is a y in R3 such that T(x)=y.

1. A. For every y in R3, there is a x in R3 such that T(x)=y. 2. B. For every y in R3, there is at most one x in R3 such that T(x)=y. 3. D. For every x in R3, there is a y in R3 such that T(x)=y.

Let A be an m×n matrix with associated transformation T(x)=Ax. 1. The row-echelon form of A has a pivot in every column. 2. The row-echelon form of A has a row of zeros. 3. Every row in the row-echelon form of A has a pivot. 4. Two rows in the row-echelon form of A do not have pivots. A. T is onto. B. T is not onto. C. There is not enough information to tell.

1. C. There is not enough information to tell. 2. B. T is not onto. 3. A. T is onto. 4. B. T is not onto.

Let T be a matrix transformation from Rn to Rm . Determine whether or not T is one-to-one in each of the following situations: 1. n=m 2. n<m 3. n>m A. T is a one-to-one transformation. B. T is not a one-to-one transformation. C. There is not enough information to tell.

1. C. There is not enough information to tell. 2. C. There is not enough information to tell. 3. B. T is not a one-to-one transformation.

What sets are linearly independant 2 -7 4 -8 -6 -12 1 pi 0 2 root(2) 0 4 sin2 0 3 e 0 2 4 1 -1 1 0 5 0 0

2 4 1 -1 1 0 5 0 0 (pivot in every row when reduced)

Suppose that A is a 5×8 matrix that has an echelon form with one zero row. Find the dimension of the column space of A, and the dimension of the null space of A. The dimension of the column space of A is The dimension of the null space of A is

4 4 The dimension of the column space is the number of pivots in the row reduced echelon form of the matrix, because the columns in the original matrix that correspond to these pivots form a basis for the column space. Since one row of the matrix A in its echelon form is zero, there are 5−1=4 pivots. Thus the dimension of the column space is 4. Since the echelon form of A has 4 columns with pivots, the solution to the homogeneous equation Ax=0 must therefore have 8−4=4 free variables corresponding to the columns that does not have a pivot. This means that the null space has a basis with 4 vectors, so its dimension is 4.

the columns of a matrix A are linearly independent if and only if

A has a pivot in every column

Determine which of the following transformations are linear transformations. A. The transformation T defined by T(x1,x2,x3)=(x1,0,x3) B. The transformation T defined by T(x1,x2)=(2x1−3x2,x1+4,5x2) C. The transformation T defined by T(x1,x2)=(4x1−2x2,3|x2|) D. The transformation T defined by T(x1,x2,x3)=(1,x2,x3) E. The transformation T defined by T(x1,x2,x3)=(x1,x2,−x3)

A. The transformation T defined by T(x1,x2,x3)=(x1,0,x3) E. The transformation T defined by T(x1,x2,x3)=(x1,x2,−x3)

Is Nul A closed under addition?

Au=0, Av=0. Must u+v be in Nul A A(u+v) = Au+Av = 0+) ---Yes

columns of matrix A are linearly independent if and only if

Ax=b has either no solution or exactly one solution (possibly depends on b)

Let T be a one-to-one matrix transformation from Rn to Rm. What can one say about the relationship between m and n? A. n<m B. n>m C. n≤m D. n≥m E. There is not enough information to tell

C. n≤m

Let A be a matrix with linearly independent columns. Which one of these statements must be true? A. The equation Ax=b always has a solution for all b. B. The equation Ax=b has a solution for all b precisely when it has more columns than rows. C. The equation Ax=b has a solution for all b precisely when it has more rows than columns. D. The equation Ax=b has a solution for all b precisely when it is a square matrix. E. The equation Ax=b never has a solution for all b. F. There is no easy way to tell if Ax=b has a solution for all b.

D. The equation Ax=b has a solution for all b precisely when it is a square matrix.

Let A be a matrix with more rows than columns. Which one of the following statements must be true? A. The columns of A must be linearly independent. B. The columns of A are linearly independent, as long as they do not include the zero vector. C. The columns of A must be linearly dependent. D. The columns of A are linearly independent, as long as no column is a scalar multiple of another column in A E. The columns of A could be either linearly dependent or linearly independent.

E. The columns of A could be either linearly dependent or linearly independent.

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for the column space of A.

F

If S is a set of linearly dependent vectors, then every vector in S can be written as a linear combination of the other vectors in S.

F

Let T: R2→R2 be the function given by T x |x| y = |y| T is a linear transformation

F

Let T: R2→R2 be the function given by T x |x| y = |y| T(cv) = cT(v) for all vectors v and scalars c.

F

Let V be the subset of R3 consisting of the vectors a b c with abc=0. is a subspace of R3

F

The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution.

F

If a set S of vectors contains fewer vectors than there are entries in the vectors, then the set must be linearly independent.

False

T: R^n --- R^m

R^n domain of T (all possible inputs) R^m codomain of T Range(T)= {T(x), x in R^n} -- lives in R^m

Is Nul A closed under scalar multiplication

Suppose u is in Nul (so Au=0) and c is any scalar Must cu be in Nul A A(cu) = cAu= c0-0 ---yes

Any set of n linearly independent vectors in Rn is a basis for Rn.

T

Let V be the subset of R3 consisting of the vectors a b c with abc=0. V contains the zero vector.

T

Let V be the subset of R3 consisting of the vectors a b c with abc=0. V is closed under scalar multiplication, meaning that if u is in V and c is a real number then cu is in V.

T

The column space of an m×n matrix is a subspace of Rm.

T

The columns of a matrix with dimensions m×n, where m<n, must be linearly dependent.

T

The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rn.

T

Two vectors are linearly dependent if and only if they are collinear.

T

any set of 4 vectors in R^3 must be linearly dependent

T

the pivot columns of a matrix are linearly independent

T

a "wide" matrix cannot have linearly independent columns

T 3x4 4 columns, max 3 pivots, columns are linearly dependent

transformation

T from R^n to R^m is a rule that assigns to each x in R^n exactly one vector T(x) in R^m. (for each input, you get exactly one output)

A mxn matrix T: R^N --- R^M given by T(x)=Ax the following are equivalent

T is one to one for each b in R^m, T(x) = b has at most one solution for each b in R^m Ax=b has no solution or exactly one solution Ax=0 has exactly one solution (the trivial solution) the column of A are linearly independent A has a pivot i every column ( this requires m>n) T is onto for each b in R^m, T(x)= b and Ax=0 are consistent consistent A has a pivot in every row

basis

a basis of V is a set {v1...vp} of vectors in v so that 1. {v1,vp} linearly independent 2. span{v1,vp} =V p= number of vector in basis for V = dimension of V A subspace usually has infinitely choices for a basis for V, but every basis for V has the same number of vectors

linearly independent

a set of vectors v1, ..., vp in R^n is linearly independent if the vector equation x1v1 + x2v2 + ... xpvp = 0 has only the trivial solution x1=...= xp=0 otherwise we say v1...vp is linearly dependent

{v1...vp} is linearly dependent if and only if

at least one of the vectors is in the span of the others you can remove at least one vector without shrinking the span

nullify A

dim (Null A)

rank A =

dim Col A

spans and subspaces

every span of vectors is a subspace and every subspace is a span of vectors

onto

f: R --- R onto if Range = R T is onto if for each b in R^m, T(X) = b has at least one solution we can always solve this equation

one to one function

function where function each output = exactly one input if for each b in R^m, the equation T(x) = b has at most one solution

{v1...vp} in R^n is linearly independent if and only if x1v1 ... + xpvp = 0

has only the trivial solution (x1=0, xp=0) Ax=0 has only the trivial solution x=0 Ax=0 has no free variables A has a pivot in every row

1 0 0 1

identity transformation

Rank theorem

if A is an min matrix: film(Col A) + dim(Nul A) = number of pivot columns f A + number of free variables of solutions set A=0 (number of non pivot columns) = n (number of columns A)

sometimes {u,v} is just a point or a line

linearly dependent

sometimes Span{u,v} is a plane

linearly independent

Let A be a 2×3 matrix. What must m and n be if we define the transformation T:Rm→Rn by T(x)=Ax? m= n=

m=3 n=2

Rank(A) + nullify(A)

number of columns of A

A subspace V of R^n is a subset of R^n satisfying

origin (0) is in v1 if u, v in v, then u + v is in V (closed under addition) if u in v and c scalar, then cu is in c (closed unfair scalar multiplication)

basis for column A

pivot column in original matrix A

1 0 0 0

projection onto x-axis

0 0 0 1

projection onto y axis

1 0 0 -1

reflects across x-axis

-1 0 0 1

reflects across y-axis

0 1 1 0

reflects across y=x

0 1 -1 0

rotates clockwise

0 -1 1 0

rotates counterclockwise

{v1, v2, v3} linearly independent if and only if

span {v1, v2, v3} is 3D

{v1, v2} linearly independent if and only if

span {v1,v2} is a plane

subspaces

study things like R^n the xy plane in R^3---- not equal to R^2 but is like R^2 line through origin in R^10 is like R but is not equal to R R^N itself

If v= span {v1...vp}, we say that V is the

subspace spanned by v1...vp (call v1...vp a spanning set for v)

basis theorem

suppose V is a subspace and dim(V) = m 1. any m linearly independent vectors in A from a basis for V 2. any m vectors that span V from a basis for V

{v1} linearly independent if and only if

the span {v1} is a line x1v1=0 has only the trivial solution x1=0 and v1 is not the hero vector

How do we find a spanning set for column A

use pivot columns in original matrix A

how do we find a spanning set for Nul A?

write solution set in parametric vector form

Is 0 in Nul A?

yes, x=0 is a solution to Ax=0

increasing span criteria

{v1...vp} is linearly independent if and only if every time you add one of these vectors into this set, the span gets larger