Geometrically, if lambda is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to lambda, then multiplying x by A produces a vector lambda*x parallel to x

Geometrically, if lambda is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to lambda, then multiplying x by A produces a vector lambda*x parallel to x

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The nullspace of a matrix A is also called the solution space of A

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if an mxn matrix A is row-equivalent to an mxn matrix B then the row space of A is equivalent to the row space of B

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The 0 vector

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is never an eigen vector, the eigen value can be 0

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If A and B are similar nxn matrices, then they always have the same characteristic polynomial equation

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The scalar lambda is an eigenvalue of an nxn matrix A when there existsa vector x such that Ax = lambda*x

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To find the eigenvalue(s) of an nxn matrix A, you can solve the characteristic equation det(lambda*I - A) = 0

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The column space of a matrix A is equal to the row space of At

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for unique eigen values,

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the eigen vectors are linearly independent

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If an mxn matrix B can be obtained from elementary row operations on an mxn matrix A, then the column space of B is equal to the column space of A

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For any 4x1 matrix X, the coordinate matrix [X]s relative to the standard basis for M4,1 is equal to X itself

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Similar matrices

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are matrices that have the same eigen values

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The dot product is the only inner product that can be defined in Rn

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A nonzero vector in an inner product can have a norm of zero

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The zero vector 0 in Rn is defined as the additive inverse of a vector

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If A is an nxnmatrix with an eigenvalue lambda, then the set of all eigenvectors of lambda is a subspace of Rn

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for every eigen value there is

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a corresponding eigen space, the basis of which is the set of eigen vectors associated with the eigen value

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If dim(V) = n, then there exists a set of n-1 vectors in V that will span V

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If dim(V) = n, then any set of n-1 vectors in V must be linearly independent

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If dim(V) = n, then there exists a set of n+1 vectors in V that will span V

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The coordinate matrix of p = 5x^2+x-3 relative to the standard basis for P2 is [p]s = [5 1 -3]T

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If P is the transistion matrix from a basis B to B', then the equation P[x]B'=[x]B represents the change of basis from B to B'

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The system of linear equations Ax=b is inconsistent if and only if b is in the column space of A

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Any diagonal entry in a diagonal or triangular matrix

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is an eigen value

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The vector -v is called the additive identity of v

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The fact that an nxn matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable

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If dim(V)=n, then any set of n+1 vectors in V must be linearly dependent

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Two vectors in Rn are equal if and only if their corresponding components are equal

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To perform the change of basis from a nonstandard basis B' to the standard basis B, the transition matrix P-1 is simply B'

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If A is an mxn matrix of rank r, then the dimension of the solution space of Ax = 0 is m-r

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the nullspace of a matrix A is the solution space of the homogeneous system Ax=0

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If an nxn matrix A is dagonalizable, then it must have n distinct eigenvalues

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If A is a diagonalizable matrix, then it has n linearly independent eigenvectors

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To subtract two vectors in Rn, subtract their corresponding components