Sample Exam Practicals

Question 1

How do I determine linear independence of a set of vectors?

Answer 1

No vector can be written as linear combination of the others.

If number of pivots in REF = number of vectors. det != 0

Question 2

How do I determine if it is a spanning set (of e.g. R^3)?

Answer 2

Any vector can be reached by a linear combination of the vectors.

If number of pivots in REF = 3.

Question 3

How do I determine if it is a basis (of e.g. R^3)?

Answer 3

If it is both linearly independent and a spanning set of R^3.

Question 4

How do I determine the dimension of Span(S)?

Answer 4

Number of linearly independent vectors.

The number of pivots in REF. (Rank)

Question 5

How do I determine a basis of the row space of A?

Answer 5

Reduce to REF, Put non zero rows, in a set.

Question 6

How do I determine a basis of the column space of A?

Answer 6

Reduce to REF, whatever columns contain pivot in REF, put corresponding columns in original matrix in a set.

Question 7

How do I determine a basis of the row space that consists of rows of A?

Answer 7

Take A transpose, find column space A^T. This will be basis of row space of A.

Question 8

How do I determine a basis of the null space N(A)?

Answer 8

Bring matrix to REF, find general solution. Abstracting free variables leads to basis.

Question 9

How do I find the eigenvalues of A? How do I find their algebraic multiplicity?

Answer 9

Solve det(A - lambda * I) = 0

Number of times they appear as solution.

Question 10

How do I find the eigenspaces of each eigenvalue? What are eigenvector and how do I find them? How do I find the geometric multiplicity of the eigenvalues?

Answer 10

Nullspace of (A - lambda * I) v = 0.

The set of eigenvectors is the basis of the eigenspace.

Nullity of A - lambda I

Question 11

If nullity of A = 0? What is the nullspace? What is the rowspace of A in terms of the rows of A?

Answer 11

Empty set.
All the rows.

Question 12

T or F. If there are more columns than rows, it cannot be linearly independent.

Answer 12

True

Question 13

Diagonalizable algorithm.

Answer 13

1. E-values must be in F
2. E-Values must be distinct or the sum of all algebraic multiplicities equals n and the algebraic = geometric for all.
3. Get eigenspaces, hence eigenbasis which form into columns of B.
4. D is eigenvalues placed in order on diagonal appearing as many times as they do in characteristic polynomial.

Question 14

Distinguish between eigenspace and basis of eigenspace.

Answer 14

Eigenspace is the nullspace of (A- lambda * I) v = 0. Basis of eigenspace is when we abstract away free variables and declare as spanning set.

Question 15

f is given function. B is input basis in function is it V or W.

Answer 15

V

Question 16

What are the steps when dealing with linear transformation?

Answer 16

Convert V and W (v.s. of bases) to be the same using isomorphism if possible.
Call basis 1 B (in V) and basis 2 C (in W)
Map B to E (in V) and map E (in W) to C
Map E (in V) to E (in W) using f.

Assuming we are looking for map from B1 to B2 using f answer is (map from E (in W) to C)* (map from E (in V) to E (in W)) * (map from B to E (in V))

Question 17

Step by step to calculate CBM?

Answer 17

M(E -> B) is the given matrix expressed in terms of the standard basis
M(E -> B)^-1 = M(B -> E).

If not easy standard basis. i.e. R2[x] = (1,x,x^2) then b1 = first equation bn = nth equation. Where b1,…,bn are the columns of M(E->B)

Question 18

Me->B, what is it?

Answer 18

CBM that takes b and outputs e.

Question 19

mat b,e (id) what is it?

Answer 19

CBM that takes b and outputs e.

Question 20

How do I calculate new coefficents in B?

Answer 20

[[v]]b (new coefficients)= CBM from old to new (Mb->e, mat e,b (id)) [[v]]g (old coefficients)

Question 21

If we have two bases in matrix form, and want to find Mb1->b2 what can we do? How would this be affected if dimF(b1) != dimF(b2)

Answer 21

Put b1 on left b2 on right. Perform ERO’s across both until b1 is I then b2 will be Mb1->b2.

If dimensions not equal there will be zero rows, we make the id as much as we can and then read off answer above zero rows on other side as our answer. The number of zero rows will be precisely the difference between the dimensions.

Question 22

When dealing with turning basis into standard basis, how do rows/columns line up?

Answer 22

row 1 = taking set to e1 of E.
row n = taking set to e2 of E.

I.e. If e1 = [1,0
0, 0] then for every 2x2 matrix in set take 1st element as row entries.

Question 23

How do function mapping work between standard basis?

Answer 23

Take 1st “column” of E i.e. e1 and insert into function, output is first column of answer. Keep going for all en.

Question 24

In the M form and mat form which one can we not find in linear transformation question directly without finding inverse?

Answer 24

M basis -> standard
mat standard,basis(idv)

Question 25

What does det(A) != 0 imply?

Answer 25

A is invertible and Ax = 0 has only the trivial solution.

Question 26

What is the formula for changing co-ordinates?

Answer 26

new co-ordinates = mat old,new(id) * old co-ordinates

Question 27

What is matv,w(f^-1)?

Answer 27

(matv,w(f))^-1

Question 28

Define L(subF)(V, W).

Answer 28

The set of all linear transformations f: V -> W.

Question 29

T or F. The dimension of L(sub F)(V, W) is dimFV * dimFW?

Answer 29

True.

Question 30

T or F. Ker(f) is isomorphic to the null space of matv,w(f).

Answer 30

True.

Question 31

T or F. Im(f) = Column Space of matv,w(f).

Answer 31

False, it is isomorphic

Question 32

Let V and V' be bases of V, and W and W' be bases of W. T or F. Nullity (mat v,w (f)) is isomorphic to column space (mat v'w' (f)). T or F. Nullity (mat v,w (f)) is isomorphic to nullity (mat v',w' (f)). T or F. Column space (mat v,w (f)) is isomorphic to column space (mat v',w' (f)) T or F. Column space (mat v,w (f)) is isomorphic to nullity (mat v,w (f))

Answer 32

False True True False

Question 33

Let f: V -> W be a linear transformation. Define rank of f and nullity of f.

Answer 33

Rank(f) = dim(Im (f)) = rank(mat v,w (f)) Nullity(f) = dim(Ker(f)) = nullity(mat v,w(f))

Question 34

Let f : V -> W be a linear transformation. T or F. dim V = dim Im(f) - dim Ker(f).

Answer 34

False, + not -

Question 35

Let f : V -> W be a linear transformation. T or F. V / Ker(f) is isomorphic to Im(f).

Answer 35

True.

Question 36

Let V be a finite dimensional vector space and let U be a subspace of V. T or F. dim (V / U) = dim V - dim U

Answer 36

True.