-
Find if invertible, then find inverse (2x2 matrix)
det(A)= ad-bc
A⁻¹= 1/det(A)*[Matrix] (In the matrix swap places of a and d. Change the sign for b an c)
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Find if invertible, then find inverse (3x3)
- 1. Find det(A)= 2[Matrix] - 4[Matrix] + 1[Matrix]
- = Anything nonzero is invertible (Lets say it is 4)
- 2. Take det(A) of all matrices and make sure the middle X values are positive and the others are negative
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| [Matrix] | -[Matrix] | [Matrix] | | [-Matrix] | [Matrix] | -[Matrix] | | [Matrix] | [-Matrix] | [Matrix] |
- 3. Find Adj(A)
- Take the top 3 values from the last step and put them on the left most columns (going down now instead of left to right) and fill out.
- -->
- 4. Find A⁻¹
- A⁻¹= 1/det(A) [Adj(A) Matrix]
Solve.
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Find all matrices that commute with the given matrix A
Set equal to each other
Solve for a,b,c,d
-
Find the matrix of a rotation through an angle of 60° in the counterclockwise direction
- Clockwise: R(-θ)= [cosθ sinθ]
- [-sinθ cosθ]
- Counterclockwise: R(θ)= [cosθ -sinθ]
- [sinθ cosθ]
60°= π/3 = cosθ= (1/2) and sinθ= (-√3/2)
- So, R(θ)= [cos(π/3) -sin(π/3)]
- [sin(π/3) cos(π/3)]
- = [1/2 -√3/2]
- [√3/2 1/2]
-
Clockwise vs Counterclockwise
- Clockwise: R(-θ)= [cosθ sinθ]
- [-sinθ cosθ]
- Counterclockwise: R(θ)= [cosθ -sinθ]
- [sinθ cosθ]
- Clockwise, the negative is bottom left.
- Counter CW, the negative is top right.
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Let L be the line in R^3 that consists of all scalar multiples of the vector [2 1 2] (going down, not horizontal). Find the orthogonal projection, of the vector [1 1 1] (also going down) about the line L
Solve for ProjL(v)= (v*a / a*a) * a
- a= vector youre projecting onto
- v= any other vector
- v*a= [2 1 2]*[1 1 1]
- = 2+1+2
- = 5
- a*a= [2 1 2]*[2 1 2]
- = 4+1+4
- = 9
(v*a / a*a) = 5/9 [2 1 2] (going down still)
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Find the matrices of the linear transformations from R³ to R³. Assume the transformation is linear:
The orthogonal projection onto the x-y plane
- Since the rank is 3, we have x, y, and z. Since the projection is only on the x-y plane, we can say that the z plane is 0.
- So, v= [x y 0]
- So, A= [1 0 0]
- [0 1 0]
- [0 0 0]
- Therefore, A*v= [1 0 0] [x] [x]
- [0 1 0] * [y] = [y]
- [0 0 0] [0] [0]
- So the matrix is:
- [1 0 0]
- [0 1 0]
- [0 0 0]
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Suppose the line L in R^3 contains the unit vector u=[u1,u2,u3]. find the matrix A of the linear transformation T(x)=refL(x). Give the entries of A in terms of the components u1,u2,u3 of u.
A= 2(u*u)* I2
- [u1] [u1² u1u2 u1u3]
- [u2] * [u1 u2 u3]= [u1u2 u2² u2u3]
- [u3] [u1u3 u2u3 u3² ]
- Then multiply all by 2 and subtract by I3
- [2u1² 2u1u2 2u1u3] [1 0 0]
- [2u1u2 2u2² 2u2u3] - [0 1 0]
- [2u1u3 2u2u3 2u3² ] [0 0 1]
- A= [2u1²-1 2u1u2 2u1u3]
- [2u1u2 2u2²-1 2u2u3]
- [2u1u3 2u2u3 2u3²-1]
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Consider the linear transformation T from R3 to R2 with:
T[1 0 0]= [7 11], T[0 1 0]= [6 9], and T[0 0 1]= [-13 17]. Find the matrix A of T.
(All matrices are vertical)
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Suppose v1, v1, ... , vm are arbitrary vectors in R^n. Consider the transformation from R^m to R^n given by:
[x1 ]
T|x2 | = x1*v1+x2*v2+...+xm*vm
|... |
[xm]
Is the transformation linear? If so, find its matrix A in terms of the vectors v1, v2,..., vm.
- Tx= Ax
- [x1 ]
- = [v1 v2 ... vm] * |x2 |
- |... |
- [xm]
A= [v1 v2 ... vm], Yes it is linear
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Find all solutions of the linear system
\x+2y= a|, where a & b are arbitrary constants .
|3x+5y= b|
- [1 2 a] (rref) ..... [1 0 -5a+2b]
- [3 5 b] -----> ---> [0 1 -b+3a]
-
The matrix [-0.8 -0.6]
[0.6 -0.8]
represents a rotation. Find the angle of rotation (in radians).
- R(Θ)= [-0.8 -0.6] = cosΘ= -0.8 and sinΘ= 0.6
- [0.6 -0.8]
- tan-1(sinΘ/cosΘ) + π
- = tan-1(-0.8/0.6+ π = 2.49
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Let L be the line in R³ that consists of all scalar multiples of [2 1 2] (going down). Find the refection vector [1 1 1] (going down) about the L.
Reflection of v= vₗₗ - v┴
- vₗₗ= ((v*a) / (a*a))*a
- and
- v┴= v-vₗₗ
- 1. Calculate vₗₗ
- ([1 1 1]*[1 2 1] / [2 1 2]*[2 1 2]) * [2 1 2]
- = (1+2+1) / (4+1+4)
- =5/9 * [2 1 2]
- = [10/9 5/9 10/9]
- 2. Now, calculate v┴
- v┴= v-vₗₗ
- = [1 1 1] - [10/9 5/9 10/9]
- = [1 - (10/9), 1 - (5/9), 1 - (10/9)]
- = [-1/9, 4/9, -1/9]
- 3. Finally, calculate the reflection
- v= vₗₗ - v┴
- = [10/9, 5/9, 10/9] - [-1/9, 4/9, -1/9]
- = [11/9, 1/9, 11/9]
-
Equation to find the orthogonal projection of a vector
projL(v)= (v*a / a*a) *a
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Equation to find the reflection of a vector
RefL(v)= vₗₗ - v┴
- where vₗₗ= (v*a / a*a) *a
- and
- where v┴= v-vₗₗ
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