Linear Algebra

Question 1

Linear Equation

Answer 1

An equation in the form a1x1+a2x2+…+anxn=b where n is a postive integer a1,a2,…,an,b are numbers and x1,x2,…,xn are variables.

Question 2

LInear System

Answer 2

A list of one or more linear eequations

Question 3

Solution to a linear system

Answer 3

A solution to one linear system a1x1+a1x2+…+anxn=b is a list of numbers (s1,s2,…,sn) such that a1s1+a2s2+…+ansn is equal to b. A solution to a linear system is a list of numbers that is simultaneously a solution to every equation in the system.

Question 4

Equivalent Linear Systems

Answer 4

Two linear systems with the same sets of variables and the same set of solutions.

Question 5

Inconsistent Linear Systems

Answer 5

A linear system with no solution

Question 6

Matrix

Answer 6

A rectangular array of numbers

Question 7

Coefficient matrix

Answer 7

For a linear equation with m equations and n variables, the mxn matrix that records the coefficients of the variable.

Question 8

Augmented matrix

Answer 8

For a linear system with m equations and n variables, the m x (n+1) matrix that records the coefficients of the variables and the constant on the other side of each equation.

Question 9

Elementary row operator

Answer 9

One of the following operations on a matrix: replace one row with the sum of itself and a multiple of another row, multiply all entries in a row by a fixed number, or swap two rows.

Question 10

Row equivalent matrices

Answer 10

Matrices that can be transformed into each other by a sequence of row operations.

Question 11

Leading entry

Answer 11

The first non zero entry in a given row, going left to right.

Question 12

Echelon form

Answer 12

A matrix is in echelon form it if has these properties: If a row is non-zero, then every row above it is also non-zero, the leading entry in one row is in a column to the right of the leading entry in each row above, if a row is nonzero, then every entry below its leading entry in the same column is zero.

Question 13

Reduced Echelon Form

Answer 13

A matrix in RREF has 1 as the leading entry in each nonzero row and has no other nonzero entries in the same column as a leading entry in a row.

Question 14

Pivot Position and Pivot Column

Answer 14

The location containing a leading 1 in the RREF of A

Question 15

Basic Variable and Free Variable

Answer 15

If A is the augmented matrix of a linear system in x1,x2,…,xn. xi is a basic variable if i is a pivot column of A and i is a free variable if i is not a pivot column of A.

Question 16

Span of a list of vectors v1,v2,…,vp that spans R^n

Answer 16

Is a set of all linear combinations of the vector.

Question 17

Linerly independent

Answer 17

The vectors v1,v1,…,vp than spans R^n are linearly independent when x1v1+x2v2+…+xpvp=0 only when x1=x2=…xp=0

Question 18

If A is a mxn vector

Answer 18

1. For each vector b in R^m, Ax=b has a solution.
2. A has a pivot in every row.
3. Every vector b in R^m is a linear combination of the column in A.
4. The span of column A is R^m.

Question 19

If a matrix have more columns than row

Answer 19

Then, they are linearly dependent

Question 20

How to determine linear independence

Answer 20

RREF the matrix then if the matrix has a pivot in every column, it is linearly independent.

Question 21

A single vector v is linearly independent only if

Answer 21

v is not equal to o

Question 22

A list of vectors in R^n is linearly dependent if

Answer 22

it includes a zero vector, some vector vi is a linear combination of the other vectors and if column>row.

Question 23

Domain, Codomain and Range

Answer 23

The domain is the input for the function, Codomain is a set that contains the output of the function, and Range is all possible output of the function.

Question 24

Linear Combination of vectors

Answer 24

Is the vector obtained by adding two or more vectors which are multiplied by scalar values.

Question 25

Linear Function f=R^n -> R^m

Answer 25

F is a transformation matrix where F(u+v)=F(u)+F(v) and F(cV)=cF(v) for u,v∈ R^n and c∈ R or there is an mxn matrix A such that F has the formula F(v)=av for v∈ R^n

Question 26

One to One Function f=R^n -> R^m

Answer 26

A function with the property that if f(u) = f(v) for u,v ∈ R^n then u=v which happens if and only if T(v)=Av where A is an mxn matrix and the columns of A are linearly independent meaning A has a pivot in every column.

Question 27

Onto Function f=R^n -> R^m

Answer 27

A function with the property y ∈ Y then there exist x ∈ X with f(x)=y. T is onto if and only if the span of the column of A is R^m, which happens when A has a pivot position in every row.

Question 28

Invertible function f=R^n -> R^m

Answer 28

A function such that for every y ∈ Y there is exactly one element x ∈ X such that f(x)=y

Question 29

Subspace of R^n

Answer 29

is a set H of vectors in Rn such that
1. the zero vector is in H. (0 ∈ H)
2. For every two vectors u and v, the sum u+v is also in H ( u,v ∈ H, u+v ∈ H)
3. If v ∈ H, and c ∈ R, then cV∈ H

Question 30

Basis of a subspace

Answer 30

is a linearly independent set of vectors where every vector in the subspace can be written as a linear combination of the basis vectors.

Question 31

Dimension of a subspace

Answer 31

is the number of vectors in any basis for the subspace.

Question 32

If A is an mxn matrix what is the (a) nullspace of A (b) column space of A (c)rank of A

Answer 32

(a) Nullspace is the set of vectors in R^n such that Av=0
(b) Column space is the space spanned by the column of A it is a subspace of R^m
(c) Rank is the number of pivot columns in A or the dimension of the column space Col A.

Question 33

Let A be an nxn matrix that is invertible, then,

Answer 33

1. There is an nxn matrix A^-1 such that AA^-1=A^-1A=In
2. For each b∈ R^n the equation Ax=b has a unique solution
3. RREF(A)=In