Questions and Answers
What does 'A' represent?
What is 'b' in relation to the identity matrix?
Row equivalent to nxn identity matrix
Does 'c' mean that A has more than n pivots?
False
Does 'd' signify that ax=0 has only non-trivial solutions?
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Does 'e' imply that the columns of A form a linearly independent set?
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Does 'f' mean that the transformation x -> Ax is not one to one?
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Does 'g' signify that the equation ax=b has at least one solution for every b in R^n?
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Is 'h' stating that the columns of A do not span R^n?
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Does 'i' describe the transformation x -> ax mapping R^n onto R^n?
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Does 'j' mean there does not exist an nxn matrix c such that CA = I?
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Does 'k' indicate the existence of an nxn matrix d such that AD = I?
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Is 'l' stating that A^t is not invertible?
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Does 'm' mean that the columns of A do not form a basis of R^n?
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Does 'n' state that Col A is not equal to R^n?
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Is 'o' referring to a dimension of Col A that is not equal to n?
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Does 'p' imply that the rank of A is less than n?
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Is 'q' stating that Nul A is not equal to {0}?
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Does 'r' imply that dim Nul A is greater than 0?
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Study Notes
Invertible Matrix Properties
- A matrix A is defined as invertible when there exists another matrix, denoted as C, such that the product CA equals the identity matrix I.
- The matrix must also be row equivalent to the n x n identity matrix, indicating full rank.
Pivots and Solutions
- An invertible matrix A contains n pivots, which are the leading coefficients in its row echelon form that indicate linearly independent rows.
- The equation ax=0 has only the trivial solution (x = 0), confirming the linear independence of columns.
Linear Independence and Transformation
- The columns of A form a linearly independent set, meaning no column can be expressed as a linear combination of others.
- The linear transformation defined as x -> Ax is one-to-one, implying distinct inputs map to distinct outputs.
Span and Basis
- For every vector b in R^n, the equation ax = b has at least one solution, meaning the columns of A span the entirety of R^n.
- The transformation x -> ax maps Rn onto Rn, signifying that every output is achievable from some input.
Basis, Column Space, and Dimensions
- The columns of A not only span R^n but also form a basis, indicating that they are both spanning and linearly independent.
- The column space of A, denoted as Col A, equals R^n, confirming that the set completely fills the n-dimensional space.
Dimensionality and Rank
- The dimension of the column space is given by dim Col A = n, representing its maximal dimensionality.
- The rank of the matrix A is n, indicating the maximum number of linearly independent column vectors it contains.
Null Space Properties
- The null space of A, denoted as Nul A, contains only the zero vector, indicating no non-trivial solutions exist for the homogeneous system.
- The dimension of the null space is dim Nul A = 0, reinforcing the idea that A is invertible as it lacks free variables.
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