Term
| What is the inverse of AB? of ABC? |
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Definition
(AB)-1=B-1A-1
(ABC)-1=C-1B-1A-1 |
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| If matrix A is m by n, then C(A) (the column space of A) has dimension _____ and is a subspace of _____. |
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Definition
| If matrix A is m by n, then C(A) (the column space of A) has dimension r and is a subspace of Rm. |
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| If matrix A is m by n, then C(AT) (the row space of A) has dimension _____ and is a subspace of _____. |
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Definition
| If matrix A is m by n, then C(AT) (the row space of A) has dimension r and is a subspace of Rn. |
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Term
| If matrix A is m by n, then N(A) (the nullspace of A) has dimension _____ and is a subspace of _____. |
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Definition
| matrix A is m by n, then N(A) (the nullspace of A) has dimension n-r and is a subspace of Rn. |
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Term
| If matrix A is m by n, then N(AT) (the left nullspace of A) has dimension _____ and is a subspace of _____. |
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Definition
| If matrix A is m by n, then N(AT) (the left nullspace of A) has dimension m-r and is a subspace of Rm. |
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Term
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Definition
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Term
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Definition
(AB)T = BTAT
Remember, transposes come in reverse order, as was the case with inverses. |
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Term
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Definition
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Term
| factorization of a symmetric matrix |
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Definition
| If A = AT is factored into LDU with no row exchanges, then U is exactly LT |
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Term
| definition of symmetric matrix |
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Definition
| A symmetric matrix has AT = A. This means that aji = aij. |
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Term
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Definition
N is the nullspace matrix of matrix A.
The columns of N are the special solutions to Ax = 0,
and AN = 0. |
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Term
| How many solutions can Ax = b have if A is full column rank? |
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Definition
| If r = m (square matrix), then Ax = b has exactly one solution. If r < m (tall, skinny matrix), then Ax = b will have exactly one solution if b satisfies the conditions imposed by the m-r zero rows, otherwise it will have no solution. |
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Term
| How many solutions can Ax = b have if A is full row rank? |
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Definition
| Ax = b will have 1 or infinitely many solutions. It will have 1 solution if r=n (square matrix). It will have infinitely many solutions that are combinations of the n-r special solutions if r < n (short, wide matrix). |
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Term
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Definition
| a set of independent vectors that span a space |
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Term
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Definition
| the number of linearly independent vectors that span the space |
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Term
| process for projecting onto a subspace |
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Definition
Solve the normal equations ATAx=ATb for x
then calculate p = Ax.
(Note: A is the matrix whose columns
are the vectors that span the subspace.)
(Note: The x here should be x hat)
(Question, why does the answer say "normal equations" not "normal equation"?) |
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Term
| What is the projection matrix P for projecting onto a subspace? |
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Definition
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Term
| formula for x hat for projection onto a line |
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Definition
x = aTb/aTa
The projection is ax.
(Note: x here should read "x hat.") |
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Term
| What is the formula for P, the projection matrix onto a line? |
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Definition
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Term
| If P is a projection matrix, what is P2? |
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Definition
P2=P
As a result, projecting a second time has no effect. |
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Term
| least squares formula for best-fit line |
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Definition
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Term
| What makes a set of vectors orthonormal? |
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Definition
A set of vectors is orthonormal if each vector in the set is a unit vector and if each vector is orthogonal to every other vector in the set.
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Term
| If the matrix Q has orthonormal columns, what special relationship do Q and its transpose possess? |
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Definition
QTQ=I
Further, if Q is a square matrix, then QQT=I |
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Term
If the columns of A are orthonormal, how does that make solving the normal equations ATAx=ATb easy?
(Note: x should be x hat) |
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Definition
If the columns of A are orthonormal, then ATA = I.
So x = ATb.
(Note: x should be x hat)
(Normally we assign a matrix with orthonormal columns the letter Q. I left is as A here for purposes of illustration.) |
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Term
| What is the relationship between the trace of a matrix (the sum of the n diagonal entries) and the sum of the n eigenvalues? |
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Definition
| The sum of the n eigenvalues equals the sum of the n diagonal entries of the matrix. |
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Term
| What is the relationship between the determinant of a matrix and the eigenvalues of a matrix? |
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Definition
| The product of the n eigenvalues equals the determinant of the matrix. |
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Term
| What are the relationships of the eigenvalues and eigenvectors of A2 to those of A? |
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Definition
| The eigenvectors for A2 are the same as the eigenvectors for A. The eigenvalues of A2 are the squares of the corresponding eigenvalues of A. |
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