Definitions
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| Linear Independence | Let V be a vector space and let v1, ... , vn be vectors in V. We say that v1, ... ,vn are linearly independent if: c1v1 + c2v2 + ... +cn vn = 0
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| Define the set of all possible linear combinations of the Vi's. | Let V = vector space & v1, v2, ... ,vn = vectors in V. The span of v1, ... ,vn "span {v1, ...vn}" is the subspace defined by span {v1, ..., vn} = {c1v1, c2v2, ..., cnvn : c1, c2, ... cn E R}.
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| When does vector v1, ... ,vn span the vector space V? | Vectors v1, ..., vn span the vector space V if any vector v E V can be written as a linear combinations of the vector v1, ... vn.
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| Define spanning sets. | Let V ba a vector space and v1,... vn be vectors in V. By a linear combination of the vectors v1, ... vn, we ge the form c1v1 + c2v2 + ... + cnvn E V with c1, ...cn E R.
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| What is the relationship between zero vector and subspace? | Every subspace must contain the zero vector of V. If 0 not= W then W not a subsapce of V
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| Let V be a vector space & let W c V be a non-empty subset. We say W is a subspace of V if: (2 conditions) | 1-Closure under Addition w1, w2 then w1 + w2 E W 2-Closure under Scalar Multiplication a E R, w E W then w E W
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| Define subspace. | Let V be a vector space vector. A subset W c V is said to be a subspace of V if W is closed under additiona dn scalar multiplication.
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| Define Vector Space | Meets all 10 axioms (1-closure under Addition, 2- Closure under Scalar Multiplication, ...)
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| Non-empty set of V is called a vector space if: | any 2 elements in V can be: 1- added together, i.e. x E V , y E V , then x + y E V (closure under addition) 2-Multiplied by scalar, i.e. x E V , scalar a E R or a E C then, ax E V (closure under scalar multiplication)
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| Real vector vs complect vectors | The elements of V are called vectors. If the scalar are reals, then V is a real vector space. If scalar is complex, then V is a complex vector space.
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| The adjoint matrix of an n x n matrix A is also an n x n matrix defined by: | adj(A) = [Cij]T of n x n = [C11 C12 ... C1n Cn1 Cn2 ... Cnn]transpose
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| Describe Determint process | Let A be n x n matrix. The minor Mij of A is the determint of the (n-1) x (n-1) matrix obtained by deleting row i & column j of A
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| Relationship between Inverse matrix and determint | A matrix is invertible if and only if its determint is NON-ZERO.
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| Properties of the DETERMINT: (4) | 1- If [A] has a row or column of 0, det (A) = 0 2-If [A] has 2 identifcal rows or columns, det(A) = 0. 3- If we switch 2 rows or columns in [A], det (A) is multiplied by "-". 4-If a row of [A] is a multiple of k, k can be pulled outside the determint.
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| What effect does row-elemetry operation have on a determint. | The determint remained unchanged under the row-elemtry operation Aij (k). (add rows)
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| Determints ca be computed along ... | any row or column.
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| We say that an n x n matrix A is invertible if: (formula) | A(-1) * A = In = A * A (-1)
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| A system will have infintely many solutions if: | Rank (A) < # of columns of A (# variable) # of free variable = Rank(A) - # of variables
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| Homogeneous systems always: | have at least 1 solution (0,0,0).
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| If Rank(A) = # of columns of A then, | the system has a unique solution.
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| If Rank(A)< Rank(A#) then, | the system has no solution.
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| Definition of Rank of A. | Let A be an m x n matrix. The # of non-zero rows of its row-echlen form is called the rank of A.
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| If A is an m x n matrix, the nullspace of A is defined by: | nullspace(A) = {x E R: Ax=0}
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