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20 Cards in this Set
- Front
- Back
Rank |
the rank of a matrix is the dimension of the vector space spanned by either the matrix's columns or rows. Rank(A)=dim(row(A))=dim(col(A)) |
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Nullity of (A) |
the dimension of the null space of the matrix dim(null(A)) |
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null space |
given a matrix A, all solutions for v where A*v=0 |
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kernel |
given a linear transform T, the kernel is the set of inputs that after you apply the linear transform, you end at the 0 vector |
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Properties of null spaces |
if vector B is the RREF of vector A, then row(A)=row(B) null(A)=null(B) col(A) =! col(B) most of the time |
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if T is a linear transform, what condition must be met in order for T to be one to one? |
the kernel of T must be 0 kern(T)={0}=0 |
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Properties of bases (pronounced like baysees) |
We are allowed to multiply an entry by a non zero scalar add one entry of a basis to another perform EROs on the vectors |
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True/False: All basis of a given span are the same dimension |
True.They may not be identical, but they must have the same number of vectors, each with the same dimension |
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Rank Nullity Theorem for a matrix |
Given a matrix A: rank(A)+nullity(A)=# of columns in the matrix |
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Range(A) |
the range of a given matrix is equal to the span of its columns Range(A)=col(A) |
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define hypervolume |
the "volume" of a matrix spanning more than R3 it is calculated using the determinant of the matrix |
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cofactor matrix |
In a given 3*3 matrix: | 1 2 3 | | 0 4 5 | | 1 0 6 | the 3*3 cofactor matrix looks like this | (4*6-5*0) -(0*6-5*1) (0*0-4*1) | the next entry is the determinant created by crossing out the 1st column and 2nd row, continuing to alternate between positive and negative | -(2*6-3*0) (1*6-3*1) -(1*0-2*1) | | (2*5-3*4) -(1*5-3*0) (1*4-2*0) | |
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determinant of a diagonal matrix | 3 0 0 | | 0 5 0 | |0 0 2 | |
determinants of diagonal matrices are equal to the product of their leading entries in each row. 3*5*2= 30 THIS IS ALSO TRUE FOR ANY TRIANGULAR MATRIX |
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Properties of determinants, given Matrices A and B |
a.) A is invertable if and only if det(A)=0 b.) det(A) = det(A)^T (linear transforms have no impact on the determinant of a matrix) c.) det(AB) = det(A) * det(B) d.) If A is triangular, then det(A) is the product of its diagonal entries |
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adjugate matrix |
the transpose of a cofactor matrix |
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define transpose of a matrix |
reflect matrix A over its diagonal ( \ ) write the rows of A as the columns of A |
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If matrix (A) has a rank of 0, what do we know about its possible solutions? |
If a rank(A)=0, the corresponding linear system will have only the trivial solution, therefor its columns are linearly independent |
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when are linear transformations invertable? |
linear transformations are only invertable when the domain and codomain have the same dimension |
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Steps to compute the basis of the null space of matrix A |
1.) row reduce A 2.) express the general solution to Ax=0 in vector form 3.) those vectors are a basis for A |
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Properties for bases |
(S is a subspace, U is the set of vectors in S) a.) if U is a basis for S, every vector in S can be written as a unique linear combination of U b) if U is linearly independent, and not a basis for S, then we can add some vectors from S to U in order to find a basis. All subspaces have a basis, just keep adding vectors in the subspace until you find it. c.) if U is linearly dependent, and spans S, we can remove some excess vectors to get a basis for S d.) if U has dim(S) vectors, and is either linearly independent or spans S, then U is already a basis for S e.) if U has fewer than dim(S) vectors, U cannot span S |