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40 Cards in this Set
- Front
- Back
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What is a Linear Equation? |
Is the equation that will give you a line.
Example: ax = b |
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What is a System of Linear Equations (SLE)? |
Is just the solution of several linear equations
Example: 3x+2y=1 x-y=3 |
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What is the SLE Theorem? |
A SLE has either: - one solution - no solution - infinitly many solutions |
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What is an Augmented Matrix of SLE? |
It is when we turn a linear equation into a matrix
Example: { 2x-4y=6} [ 2 -4|6] { } ➡️ [ ] { x-2y=1 } [ 1 -2 |1] |
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What is an Elementary Row Operations (ERO)? |
There are 3 kinds of ERO: - Multiplication - Exchange - Add a multiple of a row to another
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How to get a Row Echelon Form (REF) & Reduced Row Echelon Form (RREF)? |
RREF: 1. If we have rows is not of zeros, then the first non-zero entry has to be 1(called leading one LO). 2. If we have rows of zeros, they have to be at the bottom. 3. Each LO has to be at the right of the previous one. Example: [1 x x x x] [0 0 1 x x] [0 0 0 1 x] ➡️ REF: is steps 1 , 2 & 3⬅️ 4. Each LO row is the only non-zero entry into columns |
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How can you apply Gauss-Jordan Elimination? |
1. Find the left-most non-zero entry and put it in the first row. Then, make it a 1. 2. Turn all the numbers under the LO into zeros 3. Pretend the row/column of LO does not exist and go back to step 1 (stop here for REF) 4. Locate the right-most LO and turn the numbers above into zeros 5. Repeat step 4 with the LO until you reach the first row. (if you end up with a zero row at the bottom, you will have a general solution and give a particular solution out of it) |
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What is Gauss-Jordan's method? |
1. SLE ➡️ AM ➡️ RREF 2. If the system is inconcistent = no solution 3.If the system is concistent = -the unknown is with LO is called leading variable -The unknown without LO are called free variables (FV) or parameters |
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What is the Rank of a Matrix? |
The number of L.O. once in R. E. F. (or RREF) is called the rank of a Matrix. |
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What is the theorem for Free Variables (FV) ? |
1. #FV= #Unknowns - RANK 2. If #Unknowns > Rank then, infinitly many solutions. 3. If #unknowns = Rank then, one solution |
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What is a homogeneous system (HSLE)? |
-If the right hand side of SLE is just zeros (trivial solution) , we say the system is homogeneous. -If the system has non-zero solutions, we say it's not homogeneous (non-trivial solution). -Since HSLE has always at least the trivial solution, it's always consistent. |
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What is the HSLE theorem? |
A system is a HSLE if: 1. #U = Rank, there is only the trivial solution. 2. #U> Rank, there is no solution 3. #U> equations, there is always infinitly many solutions. Also, - Any solution is a Linear combination of the basic system. -the number of basic solutions is the FV |
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What is a basic solution? |
When solving a HSLE, the vectors which multiply the parameters are called basic solution. |
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What is a Linear combination of vectors? |
Taking a scalar times a number and adding it to other vectors multiplied with scalars. |
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What is a square matrix? |
A pxp matrix is a square matrix |
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What is a row matrix? |
Is a 1xp matrix |
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What is a column matrix? |
Is a px1 matrix also known as a vector |
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What is a main diagonal matrix? |
A main diagonal are the aii entries Example : a11 a22 a33 |
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How can we do a matrix addition? |
We simply add/subtract each entry of the same position. Example : [a11 a12] + [b11 b12] [a21 a22] [b21 b22] =[a11+b11 a12+b12] [a21+b21 a22+b22] |
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How can we multiply a matrix with a scalar? |
We simply multiply each entry by the scalar. Example : 7 [2 0] = [14 0 ] [1 3] [ 7 21] |
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What is a zero matrix? |
It's a nxm matrix that every entry is 0 |
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What is the transpose of a Matrix? |
It is when the order of a Matrix switches from pxq to qxp Example : [a11 a12] = [a11 a21] [a21 a22] [a12 a22] |
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How can we find out I'd a matrix is symmetric? |
-If matrix A=A(transpose) then is symmetric. -If matrix A=-A(transpose) then is schew-symmetric |
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What is the set Rn? |
Is the set of n-dimensional vector Example : 3×1 = R3 4×1= R4 |
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What is the zero vector? |
Is a vector containing only zeros. If a matrix A is multiplied by a zero vector the answer must be zero. |
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Explain vector/matrix multiplication. |
Any SLE can be written as = Ax=b A= matrix od coefficients x= vector of variables b= vector components |
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What are the properties of vector/matrix multiplication? |
- A(x+y) = Ax + Ay - k belongs to all real numbers - A(kx) = k (Ax) - (A+B)x = Ax +Bx |
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How are the specific solutions = homogeneous solutions + specific solutions? |
If x, is a solution Ax=b, then another solution x2 has the form x2=x0+x is some solution to Ax=0, this is the HSLE associated to Ax=b |
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How can we solve a matrix/matrix multiplication? |
For a product between two matrices to exist, the sizes must be compatible. Example: A p×q B q×r = AB p×r q's are what make the matrices compatible. We multiple both matrices for example : For same size: [a b] [e f] = [ a(e)+b(f) a(g)+b(h) ] [c d] [g h] [ c(e)+d(f) c(g)+d(h) ] |
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What are the properties of transpose? |
1. (At)t = A 2. (kA )t = kAt 3. (A+B)t = At + Bt 4. Tr(A) = tr(At) by definition 5. Transpose are unique 6. (AB)t = Bt + At |
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What is the Trace (tr) of a Matrix? |
Is the sum of all the diagonal entries. |
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What are the properties of a SLE? |
-A+B = B+A A(B+C) = AB+AC ➡️left distribution (B+C) A= BA+CA ➡️right distribution -k( AB) = kAB -Am × An = Am+n |
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What are the Non-properties of Matrix operation? |
- AB not= BA -If AB=0 neither A or B must be 0 -if AB=AC and A isn't zero, then B = C |
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What is an Identity matrix? |
Is the equivalent of ×1 for a matrix. It's composed by the LO of each row for the sizes. Matrix is always square. Example : I1= [1] I2= [1 0] [0 1] |
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What is a matrix inverse? |
It is how we "devide" Matrices. The inverse of a Matrix is an other matric such that they both have the same identity. Example: A-1= 1/A So, AB = I = BA If a matrix is not invertible we say it's singular. A matrix is singular if it does not give you the identity. |
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Is the inverse of a Matrix unique? |
Yes |
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What is a determinant? |
Is a way to find out if a matrix has an inverse. |
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What is a determinant? |
Is a way to find out if a matrix has an inverse. |
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How can we find the determinan of a 2×2 matrix? |
The formula: det(A) = ad-bc If det(A)= 0 then A is singular. If not then A is invertible. |
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How can we use the determinan to find a 2×2 matrix inverse? |
A-1= 1/ [d -b] (det(A)) [-c a] |
