Linear Algebra Flashcards
Define hyper-plane
More than 3 variables
Define shew lines
Lines in a 3 dimensional space that don’t intersect and are not parallel
Define homogenous system
A system of equations that all equations equal 0
All constant matrix solutions are 0 in an augmented matrix
Show a system of linear equations
a11x1 +… + a1nxn = b1
am1x1 + … + amnxn = bm
Define scaler
Multiplying a real number with a variable
What is a solution set
All the set of possible solutions
If a system of equations has at least one solution, what is it called
Consistent
How do you solve with elemental operation
Add or subtract 2 or more equations to remove variables and then back substitution
What are the elementary operations
Interchange order of equations
Multiple an equation by a non-zero number
Replace any equation with itself after adding it to a multiple of another equation
Solve using elementary operations
X + 3y + 6z = 25
2x + 6y + 14z = 58
2y + 5z = 19
X = 1
Y = 2
Z = 3
What is an augmented matrix
A matrix with both a coefficient matrix and a constant matrix separated by a line
How are elementary row operations different from elementary operations
elementary row operations are in the form of a matrix
Define leading entry
First non-zero entry of a row when going left to right
Rules followed to create row-echelon form
All non-zero rows are above any rows of zeros
Leading entry of a row is in a column to the right of the leading entry before it, and they are equal to 1
Difference between row-echelon form and reduced row-echelon form
In reduced row-echelon form, all entries above and below the leading entries are zero
What is a pivot position
Location of a leading entry in a row-echelon form
Define pivot column
Column that contains a pivot position
Create row-echelon and reduced row-echelon form of:
0 -5 -4
1 4 3
5 10 7
1 4 3
0 -5 -4
0 0 0
1 0 -1/5
0 1 4/5
0 0 0
What if all variable matrix entries and the constant matrix of an equation equal 0
Infinite solutions
What if all variable matrix entries of an equation equal 0 but the constant matrix entry isn’t zero
No possible solution
What if there are more variables than equations
Rather unlimited or zero soutions
What are the variables of a matrix not in a pivot column
A parameter
Difference in solving Gaussian elimination and gauss-jordan elimination
Gaussian elimination uses back substitution to solve variables
What is a basic variable
A variable entry that is not a parameter
What is a free variable
A variable entry that is a parameter
What is an equivalent matrix
A matrix that can be created from another matrix using elementary row operations
Why do homogenous systems have a trivial solution
The variables can all be zero
Define basic solution
Columns created using the parameters of a solution set
Find basic solutions
X + 4y + 3z = 0
3x + 12 y + 9z = 0
x1 = [-4] x2 = [-3]
[1 ] [0 ]
[0 ] [1 ]
What is a linear combination
Adding column matrixes together
V = a1X1 + … + anXn
Define rank
Rank is the number of rows with leading entries in a row-echelon for.
m x n Coefficient matrix of a homogenous system has how many parameters
n-r parameters
With a homogenous system how is there only one solution
If the systems rank equals the coefficient columns
Balance:
KOH + H3PO4 –> K3PO4 + H2O
3KOH + 1H3PO4 –> 1K3PO4 + 3H2O
In a matrix, how does (m x n) compare to (i,j) in a matrix
m and i are rows
n and j are columns
What are individual elements of a matrix called
Entries or components
What is a zero matrix
Matrix with only zeros
What matrixes can be added together
Matrixes of the same dimensions
What is the additive inverse of A
-A
What are the two types of vectors
Column and row vectors
What is the vector form of a system of equations
Column vectors with the variables being “scalars”
What is the matrix form of a system of equations
AX = B
All being matrixes
What must be true to multiple 2 matrixes
the first # of matrix column and second # of matrix rows, must be equal
What is the resulting (m x n) of multiplying 2 matrixes
the first # of matrix rows by the second # of matrix columns
Show A x B =
A=[1 2 1] B=[1 2 0]
[0 2 1] [0 3 1]
[-2 1 1]
[-1 9 3]
[-2 7 3]
When multiplying 2 matrixes. How can you find a specific (i,j) solution without solving the rest of the matrix.
multiple of the first row element of the i’th row of the first matrix with the first column element of the j’th column of the second matrix. Then continue this with every element in the i’th row of the first matrix with the the j’th column of the second matrix. Then add them all together.
define commute
two matrixes equal the same multiplied forwards or backwards
what is the transpose of a matrix
When you swap the rows and columns of a matrix or (i,j) –> (j,i)
(A^T)^T =
A
(AB)^T =
(B^T)(A^T)
(rA + sB)^T =
rA^T + sB^T
A(rB + sC) =
r(AB) + s(AC)
(B + C)A =
BA + CA
A(BC) =
(AB)C
k(A+B) =
kA + kB
k(pA) =
(kp)A
(k+p)A =
kA + pA
IA =
A
A+B =
B+A
(A+B) + C =
A + (B+C)
A+0 =
A
A+(-A) =
0
When is a matrix symmetric
When A = A^T
When is a matrix skew symmetric
When A = -A^T
What is an identity matrix
a square matrix with 1’s down the main diagonal and zeros in all other elements
Kronecker symbol is δ δij = 1 if i=j , 0 if i<>j
Why does the position of a identity matrix matter and what kind of identity does this prove an identity matrix is
AIn=A & ImA=A
multiplicative identity
When is a matrix considered invertible
If A(A^-1) = (A^-1)A = I
If AB = BA = I , what does this mean
B = A^-1
Find the inverse of [1 2 2]
[1 0 2]
[3 1 -1]
[-1/7 2/7 2/7]
[1/2 -1/2 0]
[1/14 5/14 -1/7]
What is the easiest way to solve AX=B if possible
X=(A^-1)B
(A^T)^-1 =
(A^-1)^T
(AB)^-1 =
(B^-1)(A^-1)
(A1*A2…Ak)^-1 =
(Ak^-1)…(A2^-1)(A1^-1)
(A^-1)^-1
A
I^-1 =
I
(A^k)^-1 =
(A^-1)^k
(pA)^-1 =
(1/p)A^-1