Exam 2 Vocabs

Question 1

Linear Transformation

Answer 1

a linear transformation is a mathematical operation that takes one vector and transforms it into another vector in a specific way.

Question 2

A transformation is linear if

Answer 2

1. T(u + v) = T(u) + T(v) for all u,v in the domain of T
2. T(cu) = cT(u) for all scalar c and all u in the domain T

Question 3

One-to-one

Answer 3

if each input has a unique output, and no two different inputs produce the same output.

Question 4

Onto (surjective)

Answer 4

if every element in the output (codomain) has at least one element in the input (domain) that maps to it.

Question 5

Domain

Answer 5

column, R^m -> R^n, R^n are the domain

Question 6

Codomain

Answer 6

Row, R^m -> R^n, R^m are the codomain

Question 7

Range(The image)

Answer 7

The set of all possible output vectors that the linear transformation can produce. It is a subset of the codomain. the range is the collection of all vectors in R^m

Question 8

The inverse of a Matrix

Answer 8

AA^-1 = I and A^-1A= I
Not all matrices have inverses. A matrix must be square (having the same number of rows and columns of n x n) and have full rank to have an inverse.

Question 9

Standard matrix of a linear transformation

Answer 9

T: R^n -> R^m is a matrix A such that when you multiply A by a vector v in R^n, you get the result applying the linear transformation T to v.

Question 10

Basis

Answer 10

A set of vectors that spans the entire space and is linearly independent. serve as building blocks, and any vector in the vector space can be expressed as a unique combination of these basis vector

Question 11

Rank

Answer 11

The maximum number of linearly independent rows or columns it contains.It is a measure of the “dimensionality” of the space spanned by the rows or columns of the matrix.

Question 12

Dimension

Answer 12

the number of vectors in any basis for that space. For example, if you have a three-dimensional vector space, any basis for that space will have three vectors, and the dimension of the space is 3.

Question 13

Column Space

Answer 13

The space spanned by the column vectors of a matrix

Question 14

Null Space

Answer 14

Set of vectors that, when multiplied by a matrix, give the zero vector. It forms a subspace of the vector space.

Question 15

Determinant

Answer 15

A scalar value associated with a square matrix, providing information about the matrix’s properties

Question 16

Singular

Answer 16

A matrix is singular if its determinant is zero, indicating potential issues with invertibility and solutions to systems of linear equations.(it has no inverse)