ma1513 1

Question 1

System with no solution

Answer 1

Inconsistent system

Question 2

System has at least one solution

Answer 2

Consistent system

Question 3

Two lines are parallel

Answer 3

No intersection –> system is inconsistent

Question 4

Two lines intersect at exactly one point

Answer 4

System has exactly one solution

Question 5

Two lines overlap

Answer 5

They represent the same line - system has infinitely many solutions

Question 6

Homogeneous system

Answer 6

Constant terms are zero

Question 7

Trivial solution

Answer 7

Of homogeneous system

All variables are 0

Question 8

Implication of homogeneous system having non-trivial solution

Answer 8

the system has infinitely many solutions

Question 9

A homogeneous system has more variables than equations - what is the nature of the solution?

Answer 9

infinitely many solutions

Question 10

Two planes in the xyz-space with the constant terms 0

Answer 10

The planes contain the origin —> Corresponds to the trivial solution

Question 11

Two planes intersect each other

Answer 11

Infinitely many solutions + trivial solution

Question 12

Two planes overlap completely

Answer 12

Infinitely many solutions + trivial solution

Question 13

Free parameters

Answer 13

Variables to be set as parameters
are those that correspond to non-pivot columns in the row echelon form

Question 14

When every column of the row echelon form is a pivot column, except the last column

Answer 14

Exactly one solution

Question 15

When there is at least one non-pivot column in the row echelon form other than the last
column

Answer 15

infinitely many solutions

Question 16

When the very last column of the row echelon form is a pivot column

Answer 16

no solution

Question 17

relation between a non-homogeneous system and its associated homogeneous system

Answer 17

General soln of non-homo system (A) = General soln of homo system + Particular soln of A

Question 18

scalar matrix

Answer 18

diagonal matrix w all diagonal entries the same

Question 19

Identity matrix behaves like __ in matrix multiplication

Answer 19

the number 1

Question 20

Transpose of symmetric matrix A

Answer 20

A^T = A

Question 21

(A + B)^T

Answer 21

A^T + B^T

Question 22

(AB)^T

Answer 22

B^TA^T

Question 23

Singular matrix

Answer 23

Does not have an inverse

Question 24

Non-singular matrix

Answer 24

has an inverse

Question 25

RREF of A is an identity matrix

Answer 25

A is non-singular (det != 0)

Question 26

REF of A has a zero row

Answer 26

A is singular (det = 0)

Question 27

(aA)^-1

Answer 27

(1/a)A^-1

Question 28

(A^T)^-1

Answer 28

(A^-1)^T

Question 29

(AB)^-1

Answer 29

B^-1A^-1

Question 30

If A is non-singular the solution of linear system Ax = b

Answer 30

Exactly one solution

Question 31

If A is non-singular the solution of Ax = 0

Answer 31

Only the trivial solution

Question 32

A is singular solution of Ax = b

Answer 32

No solution or infinitely many solutions

Question 33

A is singular solution of Ax = 0

Answer 33

Both trivial and non-trivial solutions

Question 34

det(cA)

Answer 34

cⁿdet(A) n is the size of the matrix

Question 35

det(AB)

Answer 35

det(A)det(B)

Question 36

det(A^T)

Answer 36

det(A)

Question 37

det(A^-1)

Answer 37

1/det(A)

Question 38

rank of a matrix A

Answer 38

let R be the REF of A no. of nonzero rows of R = no. of leading entries in R = no. of pivot columns in R

Question 39

rank(0)

Answer 39

0

Question 40

rank(I_n)

Answer 40

n

Question 41

for an m x n matrix rank is

Answer 41

rank <= min{m,n}

Question 42

An n x n matrix A is full rank

Answer 42

=> rank(A) = n => REF has n pivot columns => RREF is an identity matrix => A is non-singular => det(A) ! = 0

Question 43

n-space

Answer 43

he set or collection of all the n-vectors is called the n-space

Question 44

linear span

Answer 44

the set of all linear combinations of given vectors

Question 45

linear span is a __ of Rⁿ

Answer 45

subspace

Question 46

span{u}

Answer 46

represented by the line parallel to u and contains the origin

Question 47

span{u, v}

Answer 47

represented by the plane that contains the two vectors u and v and the origin. = all linear combinations su + tv

Question 48

row space of a matrix A

Answer 48

span{r1, r2 .. rn} != {r1, r2 .. rn}

Question 49

column space of A

Answer 49

span{ c1, c 2, c 3 } ≠ { c 1, c 2, c 3 }

Question 50

nullspace of A

Answer 50

a collection of vectors u such that when we pre-multiply it with A, the product Au = 0

Question 51

closure properties of subspace

Answer 51

A subspace V satisfies the closure properties: (i) for all u, v ∈ V, we must have u + v ∈ V. (ii) for all u ∈ V and c ∈ R, we must have cu ∈ V

Question 52

solution set of Ax = 0

Answer 52

subspace of Rⁿ solution space of Ax = 0 ONLY THE SOLUTION SET OF A HOMOGENEOUS SYSTEM GIVES A SUBSPACE OF Rⁿ

Question 53

homogeneous system has only trivial soln, the set of vectors is __

Answer 53

linearly independent

Question 53

if det(A) = 0 vectors are

Answer 53

linearly dependent

Question 54

homogeneous system has non-trivial soln, the set of vectors is __

Answer 54

linearly dependent

Question 55

if det(A) != 0 vectors are

Answer 55

linearly independent

Question 56

condition for linear dependence where u1, u2, ..., u k be a set of k vectors in Rn

Answer 56

k>n

Question 57

In R2 (or R3), two vectors u and v are linearly dependent if

Answer 57

they lie on the same line

Question 58

In R2 (or R3), three vectors u, v and w are linearly dependent if

Answer 58

they lie on the same line or plane

Question 59

Basis

Answer 59

smallest possible number of vectors that can generate all vectors in a given vector space they are linearly independent

Question 60

dimension

Answer 60

number of vectors in the basis of a space

Question 61

basis for row space

Answer 61

* the non-zero rows in any r.e.f. R of a matrix A * dim(row space of A) = rank(A)

Question 62

basis for column space

Answer 62

* the columns of A corresponding to pivot columns of ref R * dim(column space of A) = rank(A)

Question 63

relation bw dim(row space) and dim(column space)

Answer 63

dim(row space) = dim(column space)

Question 64

basis for linear span

Answer 64

* arrange the vectors vertically as columns * remove redundant vectors * perform G.E. to obtain ref * columns corresponding to the pivot columns will form the basis

Question 65

basis for solution space

Answer 65

use GE and separate parameters

Question 66

dim(solution space)

Answer 66

no. of parameters in the general solution = no. of non-pivot columns in R

Question 67

basis for nullspace

Answer 67

same as basis for solution space of Ax = 0

Question 68

Dimension theorem for matrices

Answer 68

If A is a matrix with n columns, then rank(A) + nullity(A) = n | rank(A) = no of pivot columns nullity(A) = no of non-pivot columns