Linear Algebra: Echelon Forms and Solutions

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Questions and Answers

What are the three requirements for echelon form?

  • All non-zero rows at the bottom
  • All entries in column below a pivot are non-zero
  • All non-zero rows at the top (correct)
  • Each leading entry of a row is in a column to the right of the leading entry of the row above it (correct)

What are the two requirements for a matrix to be row reduced?

  • Leading entry is zero (correct)
  • All entries below leading entries are non-zero
  • All rows are zero rows
  • Each leading 1 is the only non-zero entry in its column (correct)

Finding a parametric description of the solution set of a linear system is the same as solving the system.

False (B)

Can a system of linear equations with fewer equations than unknowns have a unique solution?

<p>No, it cannot have a unique solution.</p> Signup and view all the answers

What is the geometric relationship between vectors u, -v, and u-v?

<p>They form a parallelogram whose other vertex is at 0.</p> Signup and view all the answers

Span{u, v} contains only the line through u and the line through v and the origin.

<p>False (B)</p> Signup and view all the answers

What does the homogeneous equation Ax=0 imply?

<p>It always has at least one solution (the trivial solution).</p> Signup and view all the answers

When does a nontrivial solution exist?

<p>When there is at least one free variable.</p> Signup and view all the answers

What do we 'say' when vector p is added to vector v?

<p>We say that v is translated by p to v + p.</p> Signup and view all the answers

What are the four steps in writing a solution set in parametric vector form?

<ol> <li>Row reduce, 2. Express each basic variable in terms of any free variables, 3. Write a typical solution x as a vector whose entries depend on the free variables, 4. Decompose x into a linear combination of vectors using the free variables as parameters.</li> </ol> Signup and view all the answers

When is a matrix considered linearly independent?

<p>When only the trivial solution exists.</p> Signup and view all the answers

When are a set of 2 vectors linearly dependent?

<p>When they are multiples of each other.</p> Signup and view all the answers

If a set contains more entries in each vector, what can be said about the set?

<p>The set is linearly dependent.</p> Signup and view all the answers

If a set contains the zero vector, then what can be said about the set?

<p>The set is linearly dependent.</p> Signup and view all the answers

A homogeneous equation is always consistent.

<p>True (A)</p> Signup and view all the answers

The equation Ax=0 gives an explicit description of its solution set.

<p>False (B)</p> Signup and view all the answers

The solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the equation Ax=0.

<p>False (B)</p> Signup and view all the answers

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Study Notes

Echelon and Reduced Row Echelon Form

  • Echelon form requires that all non-zero rows are at the top, leading entries are shifted right compared to the row above, and all entries below a pivot are zero.
  • Row-reduced form requires leading entries to be 1, which must be the only non-zero entry in their columns, and can have zero rows.

Parametric Description and Solutions

  • Finding a parametric description of a linear system’s solution set is only valid if the system has at least one solution; otherwise, the statement is false.
  • An underdetermined system, having fewer equations than unknowns, cannot have a unique solution due to at least one free variable in the system.

Geometric Relationships

  • Vectors ( u, -v, ) and ( u-v ) form a parallelogram with one vertex at the origin (0).
  • Span of vectors ( {u, v} ) includes all linear combinations, not just the lines through ( u ) and ( v ).

Homogeneous Systems

  • A homogeneous equation of the form ( Ax=0 ) always has at least one solution, the trivial solution.
  • Nontrivial solutions in homogeneous equations exist when there is at least one free variable present.

Vector Translation

  • Translating vector ( v ) by point ( p ) results in the vector ( v + p ), effectively moving ( v ) in a direction parallel to the line defined by ( p ) to the origin.

Writing Solution Sets

  • To write a solution set in parametric vector form, follow these steps: row reduce the matrix, express basic variables in terms of free variables, write the typical solution as a vector, and decompose this vector into a linear combination of vectors using free variables.

Linear Independence and Dependence

  • A matrix is deemed linearly independent if only the trivial solution exists, indicating no vector can be expressed as a combination of others.
  • A set of two vectors is linearly dependent if one is a scalar multiple of the other.
  • A set is inherently linearly dependent if it contains more vectors than dimensions (more entries in each vector) or includes the zero vector.

Consistency of Homogeneous Equations

  • Homogeneous equations are always consistent, meaning they have at least the trivial solution.
  • The equation ( Ax=0 ) provides an implicit description of the solution set; explicit solutions are found through solving the equation.

Solution Sets of Non-Homogeneous Equations

  • The solution set of ( Ax=b ) may be empty; it is only valid under the condition that the equation is consistent for a particular ( b ) and there exists a specific solution vector ( p ).

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