Matrices & Applications Flashcards

Homogeneous Systems

• Defined as a system equal to the zero vector with right side entries all zero.
• Always possesses the trivial solution, may have non-trivial solutions depending on the system.

Trivial Solution

• A solution where all entries in the vector are zero.
• Important for defining homogeneity in systems.

Non-Trivial Solution

• A solution that contains at least one non-zero entry in the vector.

Linear Combination

• A vector can be formed as a combination of other vectors using scalars (k).
• Expressed as k1(v1) + k2(v2) + k3(v3) = v4 demonstrates v4 being a linear combination of v1, v2, and v3.

Linear Independence

• A set of vectors is linearly independent if the only scalars k1, k2, k3 satisfying the equation result in all scalars equal to zero.

Linear Dependence

• Occurs when the number of vectors (r) exceeds two and at least one vector can be expressed as a linear combination of others.

Transpose of a Matrix

• An operation where rows of a matrix become columns and vice versa.

Matrix Operation Rules

• Addition and subtraction require matrices to have the same dimensions.
• Multiplication requires matching inner dimensions.

Invertible Matrix Theory

• A matrix must be square to be invertible, but not all square matrices are invertible.
• Criteria include having n pivots, only trivial solutions for Ax = 0, linearly independent columns, and non-zero rank.

General Solution Finding

• Involves row reducing the matrix, rewriting the system in terms of variables, and solving for all variables using vector notation.

Particular Solution Finding

• Starts with finding the general solution and then substituting values to find specific real number solutions.

Consistency of a System

• A system is consistent if it has at least one solution (unique or infinite); determined by checking for pivots in each row.

Linear Combination Verification

• To check if b is a linear combination of columns of A, row reduce the augmented matrix [A|b] and verify Ax = b.

Homogeneous System Non-Trivial Solution

• Write the augmented matrix for the system, find the general solution, and check for particular solutions with free variables.

Matrix Solution as a Vector

• Solve the system represented by the augmented matrix [A|b], then express solution as x = {x1, x2, ..., xn}.

Geometric Description of Solutions

• Examine dependence on variables and relationship to the origin (0,0) or specific points based on particular solutions.

Linearly Independent Set Determination

• Use row reduction and analyze if all scalars equate to zero, indicating linear independence or dependence.

Image Under Linear Transformation

• To find x such that T(x) = b, show T(x) corresponds to the equation, construct [A|b], and determine solution uniqueness.

Linear Transformation from R2 to R2

• Construct matrix A from the system, then set up augmented matrix for transformation and find specific solution as a vector.

Network Flow Equations

• For each node in a network, establish balance equations where flow in equals flow out.

General Flow Pattern in Networks

• Create equations for each node and express total flow relationships, while ensuring all flows are nonnegative.

Matrix Inversion

• To find the inverse of matrix Q, set up [A|I] and row reduce to achieve [I|A⁻¹].

Determinant Calculation

• Use sign-changing and row operations to simplify the matrix to 2x2 matrices, applying the formula |2x2| = ad - bc.

Matrix Invertibility

• Assess the conditions of Invertible Matrix Theory; if any condition is met, the matrix is invertible.

Cramer’s Rule for Unique Solutions

• Substitute b in various columns to form new matrices, find their determinants, and determine unique solutions by avoiding division by zero.

Subspace Spanning by Vectors

• Check system consistency to confirm if vector y is spanned by set {v1, v2, v3}.

Column Space Verification

• To show vector v is in ColA, create the augmented matrix [A|v] and establish consistency.

Null Space Verification

• Construct an augmented matrix showing Bw = 0 and confirm it only results in the trivial solution.

Coordinate Vector Representation

• Express x as a combination of basis vectors, forming an augmented matrix to find c coefficients, ultimately writing [x]b as a vector.