the number of vectors in a basis for H - dim {0}=0 nonsingular matrix range of T linearly independent set of vectors dimension (of non-zero subspace H)

dimension (of non-zero subspace H)

Linear Algebra Flashcards: Matrices, Subspaces & Vector Spaces

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Algebra - Linear Algebra

user_kumartyv Created by 9 mon ago

Cards in this deck(24)

if A is an nxm matrix, A^T is the mxn matrix whose columns are formed from the rows of A

an mxn matrix A is invertible if there's an nxm matrix C such that AC=CA=I

non-invertible matrix

invertible matrix

any set H of vectors in Rn such that - 0 vector belongs to H - for any elements u and v in H, u+v also belongs to H (i.e. vector addition is preserved) - for any scalar c and any vector u in H, cu belongs to H (i.e. scalar multiplication is preserved)

subspace spanned by the vectors v1,...,vp

set of all linear combinations of the columns of A -- dim Col A = # of pivot columns -- basis formed from pivot columns of A

set of all solutions to the equation Ax=0; subspace -- dim Nul A = # of free variables -- basis formed from finding solution to equation and writing in parametric vector form

linearly independent set of vectors in H that spans H

the number of vectors in a basis for H - dim {0}=0

rank A = dim Col A

dim Nul A

[c1 ... cp] Suppose that set B={b1,...,bp} is a basis for subspace H. For each x in H, there exist scalars c1,...,cp such that x=c1b1+...+cpbp.

non-empty set V of objects called vectors on which are defined 2 operations (addition/scalar multiplication) subject to 10 axioms for all vectors u,v,w in V, all scalars c,d 1. u+v in V 2. v+u=u+v 3. (u+v)+w=u+(v+w) 4. 0 in V exists s.t. u+0=u 5. -u in V exists s.t. u+(-u)=0 6. cu in V 7. c(u+v)=cu+cv 8. (c+d)u=cu+du 9. (cd)u=c(du) 10. 1*u=u

rule that assigns each u in vector space V a corresponding T(u) in vector space W s.t. - T(u+v)=T(u)+T(v) - T(cu)=cT(u)

set of all u in V s.t. T(u)=0

set of all vectors in W of form T(x) for some x in V

a set of vectors {v1,...,vp} is linearly independent if only the trivial solution c1,...,cp=0 exists to the equation c1v1+...+cpvp=0

a set of vectors {v1,...,vp} is linearly dependent if there exist scalars c1,...,cp, not all 0, that satisfy the equation c1v1+...+cpvp=0

one-to-one linear transformation from vector space V onto W

set spanned by a finite number of vectors

set not spanned by a finite number of vectors (e.g P of all polynomials)

the set of all linear combinations of the row vectors of matrix A -- dim row A = dim col A -- basis formed by finding non-zero rows of echelon form of A

cij=(-1)^(i+j)*detAij

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Linear Algebra Flashcards: Matrices, Subspaces & Vector Spaces

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