Vector Spaces
This term describes a set of vectors that can be combined via addition and scalar multiplication.
What is a vector space?
This operation flips a matrix over its main diagonal.
What is the transpose?
This method solves linear systems by eliminating variables using row operations.
What is Gaussian elimination?
A scalar λλ such that Av=λvAv=λv for some nonzero vv.
What is an eigenvalue?
This theorem states that every square matrix satisfies its own characteristic equation.
What is the Cayley-Hamilton Theorem?
The number of vectors in a basis for a vector space.
What is the dimension?
A matrix with this property satisfies AT=AAT=A.
What is symmetric?
A system with no solutions is called this.
What is inconsistent?
A matrix is diagonalizable if it has enough of these.
What are linearly independent eigenvectors?
For a Markov matrix, all entries are non-negative, and columns sum to this number.
What is 1 in Markov matrix.?
Two vectors are this if their dot product is zero.
What are orthogonal vectors?
This matrix, when multiplied by the original, gives the identity matrix.
What is the inverse matrix?
The number of pivots in a matrix’s row echelon form determines this.
What is the rank?
This decomposition writes A=PDP−1A=PDP−1, where DD is diagonal.
What is eigendecomposition?
A linear transformation from a vector space to itself is called this.
What is an endomorphism (or linear operator)?
The process of converting a set of vectors into an orthogonal set.
What is the Gram-Schmidt process?
The determinant of this special matrix is always 1, and its inverse is its transpose.
What is an orthogonal matrix?
A transformation TT is linear if it satisfies these two properties.
What are additivity (T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)) and homogeneity (T(cu)=cT(u)T(cu)=cT(u))?
For a symmetric matrix, these special eigenvectors are always orthogonal.
What are eigenvectors of a symmetric matrix?
This property means a transformation preserves vector addition and scalar multiplication.
What is linearity?
his property means a set of vectors has no nontrivial linear combination equaling zero.
What is linear independence?
This decomposition writes a matrix as A=LUA=LU, where LL is lower triangular and UU is upper triangular.
What is LU decomposition?
This theorem states that for any matrix AA, rank(A)+nullity(A)=number of columnsrank(A)+nullity(A)=number of columns.
What is the Rank-Nullity Theorem?
This factorization, used in data science, breaks any matrix into UΣVTUΣVT.
What is Singular Value Decomposition (SVD)?
This linear algebra technique reduces dimensionality while preserving variance.
What is Principal Component Analysis (PCA)?