Questions and Answers
What is the defining characteristic of the Unifying Theorem regarding the transformation T?
T is __________.
onto
T is __________.
one-to-one
S is __________.
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S spans __________.
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S is a __________ for R^n.
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A is __________.
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Ax=b has a __________ solution.
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Ker(T) = null(A) = __________.
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Nullity(A) = __________.
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Col(A) = range(A) = __________.
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Row(A) = __________.
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Rank(A) = __________.
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Det(A) != __________.
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λ = __________ is not an eigenvalue of A.
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Study Notes
The Unifying Theorem
- Let S be a set of n vectors in R^n, and A be a matrix with transformation T: R^n -> R^n defined by T(x) = Ax.
- Transformation T is both onto (surjective) and one-to-one (injective), indicating a bijective relationship.
- Set S spans R^n, confirming that every vector in R^n can be expressed as a linear combination of vectors in S.
- Set S is linearly independent, establishing that no vector in S can be formed from others in the set.
- Set S acts as a basis for R^n, fulfilling both spanning and independence criteria.
- The equation Ax = b has a unique solution, affirming that every b in R^n corresponds to exactly one x.
- Matrix A is invertible, which signifies that it has an inverse A^(-1).
- The kernel of T (ker(T)) equals the null space of A (null(A)), and is {0}, confirming that the only solution to Ax = 0 is the trivial solution.
- Nullity of A is 0, indicating there are no free variables in the system.
- The column space (col(A)) equals the range of A, which is R^n, confirming full rank.
- The row space (row(A)) is also R^n, indicating the rows of A span the space.
- The rank of matrix A is n, meaning it has full rank and all n rows/columns are linearly independent.
- The determinant of A is non-zero (det(A) != 0), confirming that the matrix is full rank and invertible.
- Eigenvalue condition: λ = 0 is not an eigenvalue of matrix A, indicating no solutions for Ax = 0 beyond the trivial solution.
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Description
This quiz explores the Unifying Theorem in Linear Algebra, detailing the properties of the transformation T: R^n -> R^n defined by the matrix A. Participants will examine concepts such as linear independence, spanning sets, and the unique solvability of equations in R^n. Test your understanding of these core principles and their implications for vector spaces.
