Vector Spaces, Span and Basis, Null Space, Eigen Vectors and Inner Product

Question 1

What is cofactor expansion? Explain shortly how it works?

Answer 1

You take the first value and you multiply it by the determinant of the values that are in the rows below the row where the value is and the columns to the right of where the value is.

You take the second one and you follow the same judgement. you take the determinant of everything but the column and row that the value is on.

Same with the third one.

Then you do first computation - second one + third, for some reason that is unknown to me to this day :(

Question 2

Is the cofactor expansion efficient?

Answer 2

No, as in general, it requires n! multiplications.

Question 3

When a square matrix is in echelon form, there is a simple way to calculate the determinant. How?

Answer 3

The determinant of a matrix in echelon form is triangular so the determinant will be the product of the diagonal values.

But, there is one more thing to add. you multiply this result to (-1)ˆr, r meaning the number of interchanges you made to reach the echelon form.

Question 4

Reminder:

When a square matrix is in echelon form and one row is 0, the matrix a is not invertible. True or false?

Answer 4

True. Because an invertible square matrix in echelon form has pivot positions on the main diagonal. a pivot cannon be 0.

Question 5

When two rows in a matrix are equal, then the determinant of the matrix is __.

Answer 5

0

Question 6

The vector space is subject to 10 axioms. Name some.

Answer 6

1. the sum of u + v is in V
2. u + v = v + u
3. (u + v) + w = u + (v + w)
4. There is a zero vector 0 in V such that u + 0 = u
5. For each u in V, there is a vector -u in V such that u + (-u) = 0
6. The scalar multiple of u by c, denoted cu, is in V
7. c(u + v) = cu + cv
8. (c + d)u = cu + du
9. c(du) = (cd)u
10. 1u = u

Question 7

What is a vector space?

Answer 7

A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, addition and multiplications by scalars.

Maybe a better definition:
A vector space is a set of objects called vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars.

Question 8

A subspace of a vector space V is a subset H of V that has three properties:
…

Answer 8

1. The zero vector of V is in H
2. H is closed under vector addition. That is, for each u and v in H, the sum u + v is in H
3. H is closed under multiplication by scalars. That is, for each u in H and each scalar c, the vector cu is in H.

Question 9

The space Rˆn consists of …

Answer 9

… all column vectors v with n components

Question 10

Define span.

Answer 10

The span is the set of all linear combination of v1, …, vp
and is called the subset of Rˆn spanned (or generated) by v1, …, vp.

SO, Span{v1, …, vp} is the collection of all vectors that can be written in the form

  c1 v1 + c2 v2 + ... + cp vp

Maybe watch some YT videos on span to get a better sense of what it is.

Question 11

How many vectors are in Span{}?

Answer 11

Only one. The zero vector.

Question 12

How many vectors are in Span{[2,3]} over R?

Answer 12

An infinite number

{aplha[2,3], alpha in R}

Question 13

Is the span of k vectors always k-dimensional?

Answer 13

No.

Question 14

What is the span of the following?

a. {[0,0]}
b. {[1,3], [2,6]}
c. {[1,0,0], [0,1,0], [1,1,0]}

Answer 14

a. 0 dimensional because it only has the 0 vector so {}
b. 1-dimensional because one is a multiple of the other
c. 2 - dimensional because first + second = third

Question 15

It would be useful to know what trivial and nontrivial solutions are. Click

Answer 15

The system of equation in which the determinant of the coefficient matrix is zero is called non-trivial solution.

And the system of equation in which the determinant of the coefficient matrix is not zero but the solution are x=y=z=0 is called trivial solution.

Question 16

What is linear dependence?

Answer 16

Vectors v1, …, vn are linearly dependent if the zero vector can be written as a nontrivial linear combination of the vectors:

0 = a1 v1 + … + an vn

e.g., 2[1,0,0] + 2[0,2,0] - 1[2,4,0] = [0,0,0]

Question 17

If the only linear combination that equals the 0 vector is the trivial linear combination, then v1, …, vn …

Answer 17

… are linearly independent

we had this example for linear dependency
e.g., 2[1,0,0] + 2[0,2,0] - 1[2,4,0] = [0,0,0]

if you can only multiply by 0 all the vectors to get the zero vector, then the vectors v1, …, vn are linearly independent.

Question 18

For which 2-vector v over R does Span{v} consists of a finite number of vectors?

Answer 18

The [0,0] vector

Any other vector would have infinite span.

Question 19

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

Answer 19

dependent.

e. g., [2 3], [3 4], [7 8] are linearly dependent

Question 20

If the set {v1, …, vp} in Rˆn contains the zero vector, then the set is linearly ____.

Answer 20

dependent.

Question 21

H is a subspace of a vector space V. A set of vectors B = {b1, …, bp} in V is a basis for H if:

Answer 21

1. B is a linearly independent set, and
2. the subspace spanned by B coincides with H

that is, H = SpanB

Question 22

What is the column space of M?

Answer 22

The column space is the vector space spanned by the columns of M.

Question 23

What is the row space of M?

Answer 23

The row space is the vector space spanned by the rows of M.

Question 24

What is the null space of M?

Answer 24

The null space of M is the set of all solutions of the homogeneous equation Mx=0, that is the set of x’s.

Question 25

What is the dimension of a nonzero subspace H? (dimH)

Answer 25

The dimension of a nonzero subspace H is the number of vectors in any basis for H.

Remember: THE DIMENSION OF NULL SPACE OF A MATRIX IS THE NUMBER OF FREE VARIABLES IN THE EQUATION Ax= 0.

Question 26

What is the dimension of the zero subspace {0}?

Answer 26

0

Question 27

What is the rank of a matrix?

Answer 27

The rank of the matrix is the dimension of the column space of that matrix.

Remember: The pivot columns of a matrix form a basis for the column space, so
THE RANK OF THE MATRIX IS THE NUMBER OF PIVOT COLUMNS IN THE MATRIX

Question 28

What is an eigenvector?

Answer 28

An eigenvector of a n x n matrix is a nonzero vector x such that Ax = qx, for some scalar q.

The scalar is called an eigenvalue of the matrix if there is nontrivial solution to x of Ax = qx; such x is called eigenvector corresponding to q.

Question 29

Take some time now to go through the steps of calculating eigenvalues, eigenvectors and basis for the corresponding eigenspace.

Answer 29

Are you sure you took enough time?

Question 30

The eigenvalues of a triangular matrix are …

Answer 30

… the entries on its main diagonal.

BTW: A triangular matrix is a matrix that has only 0s under the main diagonal.

Question 31

A square matrix is invertible if and only if it does not have a zero eigenvalue. Why?

Answer 31

(I think) it is because when you bring a square matrix to a triangular form, then the eigenvalues are on the main diagonal. At the same time, when calculating the determinant of a triangular matrix, you compute the product of the values on the main diagonal.

If there is a 0 eigenvalue on the main diagonal, then the determinant is 0, which makes the matrix non-invertible.

Question 32

If v1, …, vr are eigenvectors that correspond to distinct eigenvalues q1, …, qn, then the set {v1, …, vn} is linearly ____.

Answer 32

independent

Question 33

Let A be an n x n matrix. Then A is invertible if and only if:

a. the number 0 is not an eigenvalue of A
b. _____

Answer 33

The determinant of A is not 0.

Question 34

What is inner product?

Answer 34

It’s dot product.

Question 35

How do you calculate the length of a vector?

Answer 35

||v|| = sqrt(v v) = sqrt(v1ˆ2 +…+ vnˆ2)

It’s based on the Pythagorean Theorem, and it makes more sense when you think of that. Google an example, the pictures are pretty self-explanatory.

Question 36

When are two vectors orthogonal?

Answer 36

When u * v = 0

There is this very long example in the slides of Lectures 13 + 14. Check it out.