Linear Algebra

Question 1

What does it mean for a matrix A to be symmetric?

Answer 1

A = A’

Question 2

A’ + B’ = ?

Answer 2

(A+B)’

Question 3

What is another name for dot product and what is it?

Answer 3

Inner product and you multiply each element of a vector with the other vector and then sum the results

Question 4

A(BC) =
A(B+C) =
(A+B)C = 
(AB)' =
(x1+x2)'y =

Answer 4

= (AB)C
= AB+ AC
= AC + BC
= B'A'
= x1'y + x2'y

Question 5

What is the trace of a matrix? [ TRACE(A) = TR(A) ]

Answer 5

The sum of the diagonal entries.

Question 6

(AB)^-1 = ?

Answer 6

B^-1 * A^-1

Question 7

det(A^-1) = ?

Answer 7

1/det(A)

Question 8

When are two vectors orthogonal?

Answer 8

When their dot product is 0

Question 9

What is a span

Answer 9

The set of all linear combinations of x1 to xn

Question 10

What is the norm of a vector (Euclidean length)?
||x|| = ?
||x||^2 =?

Answer 10

||x|| = Sqrt(sum i to n (x_i) ^2)
||x|| = sqrt(x'x)
||x||^2 = x'x

Question 11

Norm of a matrix A?

||A|| = ?

Answer 11

sqrt(sum i to n, sum j to m, x_ij^2)

Question 12

Quadratic form for a vector x associated with a matrix A?

Answer 12

x’Ax

Question 13

When is a matrix A positive definite?
When is a matrix A negative definite?
When is a matrix A positive semi-definite?
When is a matrix A negative semi-definite?

Answer 13

x’Ax > 0
x’Ax < 0
x’Ax >= 0
x’Ax <= 0

Question 14

What is the Range of an m by n matrix A?

Answer 14

The span of the n columns of A

Question 15

What is the Rank of a matrix A

Answer 15

The number of linearly independent columns

Question 16

What is the null space of a m by n matrix A?

Answer 16

the set of all n-dimensional vectors that equal the n-dimensional zero vector (the vector where every entry is 0) when multiplied by A

Question 17

What nullity of a matrix A

Answer 17

The dimension of the nullspace of A

Question 18

What is the rank nullity theorem?

Answer 18

Rank(A) + Nullity(A) = n

Question 19

What is the projection of a vector x onto the vector space J?

Answer 19

It is the vector v in space J that minimized |x-v|

Question 20

Eigenvector of a matrix

Eigenvalue

Answer 20

An eigenvector of a matrix A is a vector whose product when multiplied by the matrix is a scalar multiple of itself.

The corresponding multiplier is often denoted as lambda and referred to as an eigenvalue.

In other words, if A is a matrix, v is a eigenvector of A, and lambda is the corresponding eigenvalue, then Av = Lambda*v.

Question 21

What is a basis of a vector space V?

Answer 21

A spanning set of vectors x1 to xk that is also linearly independent.

Question 22

What is the column space?

Answer 22

The column space is the space spanned by the columns of a matrix A. The rank of A is equal to the number of linearly independent columns of A.

Question 23

What does it mean for a matrix to be nonsingular?

Answer 23

The matrix has an inverse

Question 24

What is and how do we calculate it for vectors x and y?

Answer 24

That is suggesting the inner product of x and y and we calculate the inner product by doing x’y or y’x since they are vectors.

Question 25

What is the Euclidean distance between vectors x and y ?

Answer 25

||x - y||

Question 26

What is an orthonormal basis?

Answer 26

Two vectors are said to be orthonormal if they are orthogonal to one another and each has length one

Question 27

What does it mean for a matrix P to be idempotent?

Answer 27

PP = P

Question 28

What is consistency?

Answer 28

If a system has one or more solutions vectors then it is said to be consistent. Systems without an solutions are said to be inconsistent.