Questions and Answers
What is a homogeneous system?
What defines a trivial solution?
A solution where all entries are zero.
What characterizes a non-trivial solution?
A solution that has non-zero entries.
A linear combination is expressed as ___ = k1(v1) + k2(v2) + k3(v3).
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A set of vectors is linearly independent if all scalars equal 0.
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What indicates linear dependence in a matrix?
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What must be true for matrix operations of addition and multiplication?
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An nxn matrix is invertible if it has 0 as an eigenvalue.
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How do you find a general solution?
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What is required to find a particular solution?
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A system is consistent if it has a unique or infinite solutions.
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What should be done to determine if b is a linear combination of vectors formed from matrix A?
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A homogeneous system has a non-trivial solution if it has a free variable.
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What is the procedure to describe the solution set geometrically?
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How can you determine if the columns of a matrix form a linearly independent set?
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What is the first step when T defines a linear transformation?
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To find the largest possible value of ___, you need to create flow equations for each node.
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What is the first step to find the inverse of a matrix?
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The matrix is invertible if any aspects of the Invertible Matrix Theory are true.
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What is Cramer's Rule used for?
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A system is consistent if y is the subspace spanned by vectors.
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What is the procedure to show that a vector v is in ColA?
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What does it mean if Bw = 0 has only the trivial solution?
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How can you find the coordinate vector [x]b of x?
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Study Notes
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.
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