LA Lecture 3

Question 1

How many components are there in R^n

Answer 1

n components

Question 2

If 𝑣 and 𝑤 are in a vector space S, every combination ______ must be
in S.

Answer 2

cv + dw

Question 3

One-point space Z consists of _______

Answer 3

x = 0

Question 4

y = 3x is in R^___ space.

Answer 4

R^2

Question 5

Column space of A contains _________.

Answer 5

all combinations of the columns of
A

Question 6

Ax = b is solvable when _______.

Answer 6

b is in C(A)

Question 7

Column space is a subspace of ________.

Answer 7

R^m

Question 8

[4
𝜋] is in ______ space.

Answer 8

R^2

Question 9

(1, 1,0,1,1) is in _____ space.

Answer 9

R^5

Question 10

[ 1+ i
1 - i ] is in _____ space.

Answer 10

C^2

Question 11

What are the requirements for to satisfy to be subspace of a vector space.

Answer 11

1. v + w is in subspace
2. cv is in subspace

Question 12

Does the subspace of vector include 0?

Answer 12

yes. th.e subspace of vector include 0

Question 13

Null space is subspace of _________

Answer 13

R^n

Question 14

Null space consists all combinations of special solution to _______.

Answer 14

Ax = 0

Question 15

Row space is subspace of _____.

Answer 15

R^n

Question 16

For an m by n matrix, the number of pivots variables plus the number of free variables is ______.

Answer 16

n
Counting Theorem: r + (n-r) = n

Question 17

A square matrix has no free variables. (true/false)

Answer 17

False

Question 18

An invertible matrix has no free variables.

Answer 18

True

Question 19

An m by n matrix has no more than n pivot variables.

Answer 19

True

Question 20

An m by n matrix has no more than m pivot variables.

Answer 20

True

Question 22

What is the only solution to 𝐴𝑥 = 0 if the columns of 𝐴 are independent?

Answer 22

𝑥 = 𝑜

Question 23

What is the nullspace of a matrix denoted as?

Answer 23

𝑍

Question 24

What does it mean for vectors to be independent?

Answer 24

The only zero combination 𝑐1𝑣1 + ⋯+ 𝑐𝑘𝑣𝑘 = 0 has all 𝑐’𝑠 = 0.

Question 25

If a matrix has 𝑚 < 𝑛, what can be said about its columns?

Answer 25

The columns are dependent with at least 𝑛 − 𝑚 free variables.

Question 26

When do the vectors 𝑣1,…,𝑣𝑘 span a space S?

Answer 26

If S = all combinations of the v’s.

Question 27

What are the conditions for vectors 𝑣1,…,𝑣𝑘 to be a basis for S?

Answer 27

They are independent and they span S.

Question 28

How is the dimension of a space S defined?

Answer 28

It is the number of vectors in every basis for S.

Question 29

If 𝐴 is a 4 by 4 invertible matrix, what can be said about its columns?

Answer 29

Its columns are a basis for 𝑅4.

Question 30

What is the dimension of 𝑅4?

Answer 30

4

Question 31

What is the definition of linear independence for the columns of A?

Answer 31

The only solution to 𝐴𝑥 = 0 is 𝑥 = 0.

Question 32

If three vectors are not in the same plane, what can be said about their independence?

Answer 32

They are independent.

Question 33

If three vectors w1, w2, w3 are in the same plane, what can be said about their dependence?

Answer 33

They are dependent.

Question 34

What is the condition for the columns of A to be independent in terms of rank?

Answer 34

The rank is 𝒓 = 𝒏.

Question 35

What does it mean if there are n pivots in a matrix?

Answer 35

There are no free variables.

Question 36

What must be true if n vectors in 𝑅𝑚 are present?

Answer 36

They must be linearly dependent if 𝑛 > 𝑚.

Question 37

What defines a set of vectors that spans a space?

Answer 37

Their linear combinations fill the space.

Question 38

What is the relationship between the columns of a matrix and its column space?

Answer 38

The columns span its column space.

Question 39

What is the definition of linear independence for a sequence of vectors 𝑣1, 𝑣2,…,𝑣𝑛?

Answer 39

The only combination that gives the zero vector is 0𝑣1 + 0𝑣2 + … + 0𝑣𝑛.

Question 40

What is the definition of the row space of a matrix?

Answer 40

The subspace of 𝑅𝑛 spanned by the rows.

Question 41

How is the row space of A represented?

Answer 41

It is 𝐶(𝐴^𝑇), the column space of 𝐴^𝑇.

Question 42

What are the two properties of a basis for a vector space?

Answer 42

* The basis vectors are linearly independent * They span the space.

Question 43

How is the dimension of a vector space defined?

Answer 43

The number of vectors in every basis.

Question 44

What is the dimension of the whole n by n matrix space?

Answer 44

𝑛²

Question 45

What is the dimension of the vector space M containing all 2 by 2 matrices?

Answer 45

4

Question 46

If columns of matrix are dependent, so are the rows. true/false

Answer 46

False

Question 47

The column space of 2 by 2 matrix is the same as its row space. true/false

Answer 47

False

Question 48

The column space of 2 by 2 matrix has the same dimension as its row space. true/false

Answer 48

True

Question 49

The columns of a matrix are a basis for column space. true/false

Answer 49

False.

Question 50

What is the dimension of the column space C(A) and the row space C(A^T)?

Answer 50

Both have dimension r (the rank of A).

Question 51

What is the dimension of the nullspace N(A)?

Answer 51

The dimension is n - r.

Question 52

What is the dimension of the left nullspace N(A^T)?

Answer 52

The dimension is m - r.

Question 53

What does elimination produce for the row space and nullspace of A?

Answer 53

It produces bases that are the same as for R.

Question 54

How does elimination affect the column space and left nullspace?

Answer 54

Elimination often changes them, but their dimensions don't change.

Question 55

What is a rank one matrix and its column space basis?

Answer 55

A rank one matrix is A = uv^T; C(A) has basis u.

Question 56

What is the basis for C(A^T) in a rank one matrix?

Answer 56

C(A^T) has basis v.

Question 57

If m=n, then the row space of A equals the column space. true/false

Answer 57

false

Question 58

The matrices A and −A share the same four subspaces. true/false

Answer 58

true

Question 59

If A and B share the same four subspaces, then A is a multiple of B. true/false

Answer 59

false