Linear Algebra Flashcards
What does it mean for a matrix A to be symmetric?
A = A’
A’ + B’ = ?
(A+B)’
What is another name for dot product and what is it?
Inner product and you multiply each element of a vector with the other vector and then sum the results
A(BC) = A(B+C) = (A+B)C = (AB)' = (x1+x2)'y =
= (AB)C = AB+ AC = AC + BC = B'A' = x1'y + x2'y
What is the trace of a matrix? [ TRACE(A) = TR(A) ]
The sum of the diagonal entries.
(AB)^-1 = ?
B^-1 * A^-1
det(A^-1) = ?
1/det(A)
When are two vectors orthogonal?
When their dot product is 0
What is a span
The set of all linear combinations of x1 to xn
What is the norm of a vector (Euclidean length)?
||x|| = ?
||x||^2 =?
||x|| = Sqrt(sum i to n (x_i) ^2) ||x|| = sqrt(x'x) ||x||^2 = x'x
Norm of a matrix A?
||A|| = ?
sqrt(sum i to n, sum j to m, x_ij^2)
Quadratic form for a vector x associated with a matrix A?
x’Ax
When is a matrix A positive definite?
When is a matrix A negative definite?
When is a matrix A positive semi-definite?
When is a matrix A negative semi-definite?
x’Ax > 0
x’Ax < 0
x’Ax >= 0
x’Ax <= 0
What is the Range of an m by n matrix A?
The span of the n columns of A
What is the Rank of a matrix A
The number of linearly independent columns
What is the null space of a m by n matrix A?
the set of all n-dimensional vectors that equal the n-dimensional zero vector (the vector where every entry is 0) when multiplied by A
What nullity of a matrix A
The dimension of the nullspace of A
What is the rank nullity theorem?
Rank(A) + Nullity(A) = n
What is the projection of a vector x onto the vector space J?
It is the vector v in space J that minimized |x-v|
Eigenvector of a matrix
Eigenvalue
An eigenvector of a matrix A is a vector whose product when multiplied by the matrix is a scalar multiple of itself.
The corresponding multiplier is often denoted as lambda and referred to as an eigenvalue.
In other words, if A is a matrix, v is a eigenvector of A, and lambda is the corresponding eigenvalue, then Av = Lambda*v.
What is a basis of a vector space V?
A spanning set of vectors x1 to xk that is also linearly independent.
What is the column space?
The column space is the space spanned by the columns of a matrix A. The rank of A is equal to the number of linearly independent columns of A.
What does it mean for a matrix to be nonsingular?
The matrix has an inverse
What is and how do we calculate it for vectors x and y?
That is suggesting the inner product of x and y and we calculate the inner product by doing x’y or y’x since they are vectors.
What is the Euclidean distance between vectors x and y ?
||x - y||
What is an orthonormal basis?
Two vectors are said to be orthonormal if they are orthogonal to one another and each has length one
What does it mean for a matrix P to be idempotent?
PP = P
What is consistency?
If a system has one or more solutions vectors then it is said to be consistent. Systems without an solutions are said to be inconsistent.
