Linear Algebra Theorems Flashcards

Questions and Answers

What does Chapter 1 Theorem 1 state?

Each matrix is row equivalent to one and only one reduced echelon matrix.

A _____________ in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A.

pivot position

What does Chapter 1 Theorem 2 - Existence and Uniqueness Theorem state?

A linear system is consistent IFF the rightmost column of the augmented matrix is not a pivot column.

What does the term 'span' refer to?

The set of all linear combinations of vectors v1,...,vp. Signup and view all the answers

What is the definition of the product of A and x?

The linear combination of the columns of A using the corresponding entries in x as weights. Signup and view all the answers

What does Chapter 1 Theorem 3 state?

The matrix equation Ax = b has the same solution set as the vector equation x1a1 + x2a2 +...+xnan = b. Signup and view all the answers

What does the existence of solutions theorem state?

The equation Ax = b has a solution IFF b is a linear combination of the columns of A. Signup and view all the answers

What is the conclusion of Chapter 1 Theorem 4?

For a particular mxn A, each b in Rm is a linear combination of the columns of A. Signup and view all the answers

What does the Row-Vector Rule for Computing Ax state?

The ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x. Signup and view all the answers

What does Chapter 1 Theorem 5 state about matrices?

A(u+v) = Au + Av and A(cu) = c(Au). Signup and view all the answers

What are homogeneous linear systems?

The homogeneous equation Ax = 0 has a nontrivial solution IFF the equation has at least 1 free variable. Signup and view all the answers

What does Chapter 1 Theorem 6 elaborate on?

If the equation Ax = b is consistent for some given b, the solution set of Ax = b is all vectors of the form w = p + vh. Signup and view all the answers

What does linear independence mean?

An indexed set of vectors is linearly independent if the vector equation has only the trivial solution. Signup and view all the answers

What does the Chapter 1 Theorem 7 characterize?

An indexed set of two or more vectors is linearly dependent if at least one vector is a combination of others. Signup and view all the answers

What does it mean if a set contains more vectors than there are entries in each vector?

The set is linearly dependent. Signup and view all the answers

What does Chapter 1 Theorem 9 state about the zero vector?

If a set contains the zero vector, the set is linearly dependent. Signup and view all the answers

What are the conditions for a transformation to be linear?

T(u+v) = T(u) + T(v) and T(cu) = cT(u) for all u,v and scalars. Signup and view all the answers

What does Chapter 2 Theorem 1 state regarding matrix addition?

A+B = B+A. Signup and view all the answers

What is the definition of multiplication in matrix terms?

Each column of AB is a linear combination of the columns of A using weights from B. Signup and view all the answers

What does invertibility mean for a matrix?

A square matrix A is invertible IFF det A ≠ 0. Signup and view all the answers

What is the formula for the inverse of a 2x2 matrix?

A^(-1) = 1/(ad-bc)[d -b -c a]. Signup and view all the answers

What does Chapter 3 Theorem 7 state about Cramer's Rule?

The unique solution x of Ax = b is given by xi = (det Ai(b))/(det A). Signup and view all the answers

What is the determinant of a triangular matrix?

The product of the entries on the main diagonal. Signup and view all the answers

Study Notes

Theorems and Definitions in Linear Algebra

Each matrix is uniquely corresponding to one reduced echelon form, establishing essential properties of matrices.
A pivot position in a matrix is associated with a leading 1 in the reduced echelon form; columns with these positions are termed pivot columns.
Existence and Uniqueness Theorem: A linear system is consistent if the rightmost column of the augmented matrix is not a pivot column, leading to either a unique solution or infinitely many solutions based on the presence of free variables.
The Span of vectors (v_1, ..., v_p) in (R^n) includes all possible linear combinations of these vectors.
The product of matrix (A) and vector (x) represents a linear combination of the columns of (A) with weights given by (x).
The matrix equation (Ax = b) shares the same solution set as the corresponding vector equation formed by the matrix's columns.
The equation (Ax = b) has solutions if and only if (b) can be formed as a linear combination of the columns of (A).
For a given matrix (A), a consistent equation (Ax = b) leads to a solution set expressed as a specific solution (p) plus solutions to the homogeneous equation (Ax = 0).
Vectors in a set are linearly independent if the only solution to the vector equation combining these vectors equals zero is the trivial solution.
A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of others in the set.
If more vectors are present in a set than there are entries in each vector, the set is guaranteed to be linearly dependent.
The presence of the zero vector in a set ensures that the set is linearly dependent.

Linear Transformations and Properties

A transformation (T) is linear if it satisfies two properties: additivity and homogeneity.
For a linear transformation (T), (T(0) = 0) is always true, and it maintains linearity through scalar multiplication and addition.
A linear transformation (T: R^n \to R^m) can be represented by a unique matrix (A) such that (T(x) = Ax) for all (x).
Multiple specific transformations are represented by matrices which define changes like reflections through axes, contractions, expansions, and projections.
A mapping is onto if every element in (R^m) has a pre-image in (R^n); it is one-to-one if no two different elements in (R^n) map to the same element in (R^m).
A linear transformation is one-to-one if the kernel is trivial (only contains the zero vector). It's onto if the columns of the transformation's matrix span (R^m).

Matrix Operations and Inverses

Matrix addition follows commutative and associative properties, alongside scalar distributive laws.
The definition of matrix multiplication involves constructing columns of the resultant matrix as linear combinations of columns from the first matrix, weighted by entries in the second.
Determinant properties highlight that for triangular matrices, the determinant is the product of the diagonal entries; and transformations like row operations impact the determinant in defined ways.
A matrix (A) is invertible if its determinant is non-zero; the relationship between matrices (AB = I) also implies their invertibility.
The adjugate matrix provides a formula for the inverse as long as the determinant is non-zero; areas and volumes defined by matrices can be calculated using the absolute value of determinants.

Concepts of Subspaces and Dimensions

A subspace in (R^n) must contain the zero vector, be closed under vector addition, and subject to scalar multiplication.
The column space of a matrix comprises all possible linear combinations of its columns, while the null space is the solution set of the homogeneous equation (Ax = 0).
Dimensions reflect the number of vectors in a basis for a subspace, impacting calculations like rank, which is the dimension of the column space.
The Rank Theorem establishes a relationship between the rank of a matrix and its nullity (dimension of the null space).
The Basis Theorem asserts that a linearly independent set of the correct size spans a subspace, making it a basis.

Determinants and Their Properties

The determinant of a matrix quantifies area or volume; properties include behavior under row operations and multiplicative relationships between the determinants of products.
Cramer's Rule connects determinants to solution entries in linear systems with the matrix's inverse reflecting the uniqueness of solutions based on non-zero determinants.
The Invertible Matrix Theorem synthesizes criteria for invertibility, pivotal in linear algebra's broader applications.

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Description

Test your understanding of key theorems and concepts in Linear Algebra with this flashcard quiz. Covering topics such as row equivalence and pivot positions, these cards are perfect for reinforcing your knowledge from Chapter 1 of Linear Algebra.