FInal Examination Flashcards
Memrize the fuck out of this information
Explain the solution
Since these two vectors are linearly independent, we can find the perpendicular basis by simply setting them to row vectors and solving for the new basis.
|1 2 3 4 | —-> C [1 -2 1 0]
|5 6 7 8 | **D [2 -3 0 1] **
Explain the solution
Find the projection onto each vector using the (dot product)/magnitudes * unit vector of each vector.
Then add up all the new projections.
Explain the Solution
₁₂Start by finding the Unit Vector for 𝓥 using:
𝓾₁ = [1 / ||𝓥₁||] * [𝓥₁]
Then we will find the perpendicular 𝓥₂ using the equation: 𝓥₂⊥ = 𝓥₂ - (𝓾₁ ⋅ 𝓥₂)𝓾₁
then we find 𝓾₂ using
𝓾₂ = [1 / ||𝓥₂⊥||] * [𝓥₂⊥]
What is equation to find 𝓥₃⊥ and 𝓾₃ using Gram–Schmidt process ?
𝓥₃⊥ = 𝓥₃ - (𝓾₁ ⋅ 𝓥₃)𝓾₁ - (𝓾₂ ⋅ 𝓥₃)𝓾₂
then: 𝓾₃ = [1/||𝓥₃⊥||] * 𝓥₃⊥
Go about solving this problem ?
Start by setting this matrix equal to 0 and solving for the Kernel.
Create a matrix out of the Kernel [-t -r r t] then set to two kernels to their own linearly independent vectors
then solve using the Gram–Schmidt process for the Orthonormal Basis
With regard to Determinants what condition is a matrix invertible ?
when det(A) = ad - bc ≠ 0
What can we conclude about the relationship between det(A) and its transposed matrix det(A^t)
det(A) = det(A^t) so long as its a square matrix
How to find Determinants of 4x4 matrix ?
set the first row as the coefficient and follow the crossout method in order to set a few rows sets of 3x3 alternating in positivity. then solve each 3x3 using the same method as before.
Explain the Solutions to these
- Since there is a scalar being applied to a row we must multiply 8 * -9 = -72
- Since a row was switched it must be (-1)¹8 = -8
- Since there is 2 row switched it must be (-1)²8 = 8
- The operation of adding another row times a scalar doesnt change the det(A) so 8
How to find Basis of the Subspace of R^3 defined by equation:
2x₁ + 3x₂ + x₃ = 0 ?
adjust the equation:
x₁ = - (3/2)x₂ - (1/2)x₃
then set x₂ = t; x₃ = s
we now have [x₁, x₂, x₃] = [- 3/2t -1/2s, t, s]
this is now the two basis t[-3, 2, 0] & s[-1, 0, 2]
How can i tell if a Vector x is a part of Span V of vectors 𝓥₁ …… 𝓥ₘ
We can set all the vectors into an augmented matrix equal to vector x and determine whether there a coordinates which can be solved for.
Solve
Convert from 2x2 format to R^4. line the vectors into columns. then solve for linear independence.
Solve to Basis
|1 1|
|0 1|
SImply set a Basis with arbitrary letters
|a b|
|0 c| Then solve a dot product between the two matrix’s
|1 2| |a b| = |a b+c |
|0 3| |0 c| |0 3c | Then align vertically for our matrix and solve for RREF
Explain Solution
Set M at each Basis
Align the new Matrix Vertically
Then solve RREF
Explain Solution
Use Diagram:
x + iy → x - iy
| |
|x| |x |
|y| |-y| Then convert to matrix and solve RREF
Find all the 2x2 Matrix such that e₁ = |1|
|0|
is an EigenVector with EigenValue of 5
Start by setting an arbitrary 2x2 matrix times e₁ = 5 e₁
|a b||1| = 5|1|
|c d||0|……..|0|
Then solve for the values of a b c d that would make the 2x2 which would satisfy the statement.
Find an EigenBasis for a Rotation 180
All we are doing is flipping the axis of a basic Identity Vector:
|-1 0|x
|0 -1|y
equation for EigenValues
We can simply use the determamite, set it to 0 and solve for a value of λ that makes the det = 0
det(A - λＩₙ) = 0
Answer all
1. Yes we can solve A³ down to λ³𝓥 through extrapolation
2. Yes it is simply 1/λ as we can basic algebra to solve for this
3. Yes it is simply λ + 2
4. Yes it is simply 7λ
How to solve for the EigenBasis ?
Start by setting the det(A - λＩₙ) = 0
det |7 8| - |λ 0| = 0
……..|0 9| |0 λ|
|7-λ 8 | → (7-λ)(9-λ) = 0
|0 9-λ|…..our EigenValues are 7 and 9
now we will solve for the Kernel with these new values:
Ker…|7 8| - |7 0|
…………..|0 9|……….|0 7|
|0 8|x = 0 → y = 0…….our first kernel = |1|
|0 2|y………………………………………………………|0|
