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44 Cards in this Set
- Front
- Back
How do you know if a matrix is singular?
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If it's Det = 0.
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What does having a singular matrix mean?
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That the matrix vectors are lin dependant.
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What is the det of a triangular matrix?
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The product of all its diagonal values.
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What is the det of the Id. matrix?
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1
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What is Det ( A -1 ) ?
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1/ Det (A)
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Det ( AB ) =
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Det ( BA )
Det (A) * Det (B) Det (B) * Det (A) |
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IF V is an n x n triangular matrix, then Det (V ^t) =
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Det (V)
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A - L*I is singular if...
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Det (A - L*I) = 0
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A scalar L is an eigenvalue of A if and only if ...
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A - L*I is singular
and Det (A - L* I) = 0 |
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If A is an n x n matrix, then Det (A - L*I) is a ....
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ploynomial of degree n in t.
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What kind of vectors will span all of R^n?
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ANY number n vectors which are not scalar multiples of each other will span all of R^n.
Cool. |
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How can you use scalars c1 through cn to produce formulas for the span of a set of vectors?
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Multiply each vector in the set by a unique scalar cn and concatenate the matrix with x, y, z, etc. then solve for each scalar to get formulas in x, y, z..
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What are the two conditions for a basis of a vector space?
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That they span that vector space.
That they are linearly independent. |
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If a set of vectors spans S, then we automatically know that it is a ....
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subspace of S.
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What is the definition of a subspace?
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Any set of vectors which satisfy the two axioms of a vector space.
Closed under addition and under scalar multiplication. |
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How do you find the eigenvalues of A^2?
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If Av = Lv
Then A^2v = A (Av) = A (Lv) = L (Av) = L (Lv) = L^2v. Just square the eigenvalue. |
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How do you find the eigenvalues of A ^ -1?
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If Av = Lv , then v = (Av) / L since L must be nonzero.
A ^-1 v = A^ -1 (Av / L) = A^ -1 A ( v / L) = v(1 / L) |
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How do you find the eigenvalues of (A ^ -1 - I) ?
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By Av = Lv
(A ^ -1 - I)v = ( 1 / L - I)v |
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What does it mean for Lamda to be an eigenvalue of A?
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It means that A - LI is singular.
When its determinant = 0 |
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How do you find the null space of a set of vectors and what is the definition?
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All vectors for which Ax = 0
Set the concatenated augmented matrix to RREF and then solve the system resulting from it. |
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What is the definition for the range of a set of vectors?
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The set of all vectors y in R^m such that Ax = y is consistent.
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K is a nonzero scalar, and kA has an inverse, what is it?
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(1 / k) A ^ -1
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What is the inverse of A ^ t?
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( A ^ -1) ^t
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If A has an inverse, then what is A*A^ -1?
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A*A ^-1 = A^ -1 * A = I
A matrix multiplied either way by its inverse = the identity matrix. |
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If the Det of a matrix = 0, what does this mean?
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That the matrix cannot have an inverse. It is singular. So, the vectors in the matrix are linearly dependent, there is no unique solution for this set of vectors.
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What is the formula for the inverse of a matrix if the determinant is not equal to 0?
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The matrix is nonsingular, the vectors are lin independent.
A^ -1 = 1/Det * [ d -b -c a ] |
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What is the idea behind solving a system of equations using its matrix and its inverse?
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By Ax = b we have x = A^-1 b
Take the coefficient matrix, find its inverse using the identity matrix, then multiply the inverse matrix by the vector b on the RHS, this will give you your variables x1, x2..... xn. You can then verify that they are correct and satisfy the original equations. |
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How can I get Q ^ -1 if Q = A^t * B^t?
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Q^ -1 = (A^ -1)^t * (B^-1)^t
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What is the formula for the angle between two vectors?
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u*v = [u][v]cos(theta)
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What the formula for the projection of vector b onto vector a?
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p = b*a / [a] * a / [a]
So, proj b onto a is just: b times a squared over length of a squared. |
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What is the strategy for finding eigenvalues?
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Multiply your matrix by Lamda times the identity matrix, take the determinant of that, set it = 0, solve for all values of lamda.
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What is the strategy for finding eigenvectors?
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Take your eigenvalues one at a time, plug them into A-L*I, then solve that homogenous system.
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If you raise L to power of some integer k, how does that relate to your original matrix?
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Raise the original matrix to the same power to retain the relationship, so, if you raise a mtrix to a power, then you know that you can raise your L values to the same power and do not need to find them again.
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If the det of A is nonzero, then it is nonsingular and the vectors of A are lin independent. Then A has an inverse, assuming that A has eigenvalues L, how do they relate to A^ -1?
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1 / L is an eigenvalue of A^ -1
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For any scalar alpha added to lamda (L + a), how does that relate to the original matrix A?
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(L + a) = ( A + a*I)
This means that if you want to add values to either your original matrix or to the found lamdas, you can still preserve your relationship between the two. |
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Do A and A^ t have the same eigenvalues?
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You betcha!!! :D
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If you happen to get a triangular matrix, what is the shortcut to finding your eigenvalues quickly?
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You eigenvalues will simply be the values of the diagonal entries.
Easy peasy. |
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What makes a matrix defective?
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If the geometric multiplicity is less than the algebraic multiplicity.
If lamda has algebraic multiplicity 2 but there is only one vector in the eigenspace, then the matrix is defective. |
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What is the formula that you want to use to diagonalize a matrix?
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You'll be wanting that the diagonal matrix D = S^ -1 * A * S
So then you get S from the eigenvectors of A. BAM!!! :D |
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What is diagonal matrix D ^10 having to do with matrices A and S such that S can diagonalize A?
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D ^10 = S^ -1 * A^10 * S
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If you have D^ 10 = S^ -1 * A^10 * S.... how can you get what just A^10 = ...?
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A^ 10 = S* D^10 & S^-1
So, just flip everything backwards and you're good. |
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What's the gameplan for diagonalizing a matrix?
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1) Find eigenvalues and form eigenvectors.
2) The eigenvectors give you the matrix you'll use to diagonalize the one that the eigenvalues came from. 3) Plug your matrices into the formula D = S^-1 * A * S 4) Ta-da!!! $$$$$$$ yeah. |
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What is the definition of an orthogonal matrix?
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Q^t * Q = I
The transpose times the original matrix equals the identity. |
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How do you find the least squares solution so an inconsistent system?
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Use the formula A^t*A*x = A^t*b
Multiply out, then you have two or three equations, solve for your x's all in terms of b, then your vector is b times whatever values it holds for the various x's. |