Linear Algebra

Question 1

When we say a matrix A is an m x n matrix, what do we mean?

Answer 1

We mean A has m rows and n columns

Question 2

How many eigenvalue eigenvector pairs does an n by n matrix have?

Answer 2

N, if you include the complex number field and count multiplicities

Question 3

What are eigenvalue eigenvector pairs?

Answer 3

They (lambda, v) are pairs associated with a particular matrix A such that Av = lambda

Question 4

So what is the general significance of eigenvectors?

Answer 4

An eigenvector is a vector for a matrix such that when multiplied by the matrix is only scaled linearly

Question 5

What is the Hessian?

Answer 5

A matrix of all second order partial derivatives of a function that has multiple variables

Question 6

Why is the Hessian a matrix and not a vector? Wouldn’t the data structure containing all second order partial derivatives of a multi variabled function just need one entry per second order partial derivative for each variable?

Answer 6

Question 7

What does simultaneously-diagonalizable mean?

Answer 7

..

Question 8

What is a diagonal matrix?

Answer 8

Question 9

What does the matrix P serve as when performing matrix diagonalisation?

Answer 9

the change-of-basis matrix that takes you from the standard basis to the new basis.

Question 10

What are the columns of the matrix P?

Answer 10

The eigenvectors of A

Question 11

A 2x3 matrix represents a linear transformation from a _-dimensional space to a _-dimensional space.

Answer 11

3;2

Question 12

What is a basis?

Answer 12

a set of linearly independent vectors that span a vector space. In simpler terms, you can represent any vector in the space as a unique linear combination of the basis vectors. The number of basis vectors is equal to the dimension of the vector space

Question 13

What is the data type of a basis

Answer 13

A set of vectors

Question 14

What does a matrix represent?

Answer 14

A linear transformation between vector spaces

Question 15

What data type do you use to represent nonlinear transformations between spaces?

Answer 15

Essentially functions

Question 16

Are linear transformations performed on vectors or a space?

Answer 16

Linear transformations are defined as mappings from one vector space to another, but they act on individual vectors within those spaces. So, in a sense, both perspectives are correct:

Question 17

What is shearing?

Answer 17

is a type of linear transformation that distorts the shape of an object along one or more axes, effectively “slanting” it. In a shear transformation, parallel lines remain parallel, but they are no longer equidistant, and angles between lines can change. However, areas or volumes are preserved under shearing.

Question 18

What does a scaling matrix look like?

Answer 18

Diagonal entries have values and all other entries are zero

Question 19

What happens when a matrix has has non diagonal entries?

Answer 19

If elements in the non diagonal are nonzero then the transformation is no longer scaling and could be rotation, shearing, reflection etc