Linear Algebra Midterm Exam Flashcards
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Questions and Answers

What is a Linear Equation?

A linear equation in 'n' variables is an equation which can be written in the form a1x1 + a2x2 + ... + anxn = b where a1 is a real number for i = 1, 2, ... , n and so is b.

What is a solution to an equation?

A solution to an equation using 'n' variables is an ordered 'n-tuple' (c1, c2, ... , cn) such that when xi is replaced with ci for all i = 1, 2, ... , n, the resulting statement is true.

What is a system of linear equations?

A system of linear equations is taken to be at least one (probably more) linear equations considered jointly. A solution to a system of linear equations is simply a solution to every equation in the system (if such a thing exists).

What does the Theorem about a System of Linear Equations state?

<p>Any system of linear equations has exactly one of the following: unique solutions, infinitely many solutions, or no solutions.</p> Signup and view all the answers

What kind of solution(s) will you have in a consistent system?

<p>Unique or infinitely many solutions</p> Signup and view all the answers

What kind of solution(s) does an inconsistent system have?

<p>No solutions.</p> Signup and view all the answers

What is a matrix?

<p>An mxn matrix is a regular array having 'm' rows and 'n' columns.</p> Signup and view all the answers

What is Scaling in Elementary Row Operation?

<p>Scale all entries of a single row by a non-zero scale.</p> Signup and view all the answers

What is Swapping in Elementary Row Operation?

<p>Interchange rows of the matrix.</p> Signup and view all the answers

What is Replacement Operation in Elementary Row Operations?

<p>Scale a single row of the matrix by a non-zero scalar and add the result to a different row, replacing that row.</p> Signup and view all the answers

What is a row equivalency?

<p>A row equivalency is matrices representing equivalent systems, also known as equivalent matrices.</p> Signup and view all the answers

When is a matrix said to be in Row Echelon Form?

<ol> <li>First non-zero leading entry in each row is to the right of the leading entry in the row above; 2. Entries below each leading entry are zeros; 3. Any rows of all zeros are at the bottom of the matrix.</li> </ol> Signup and view all the answers

What additional condition must a matrix satisfy to be in Reduced Row Echelon Form?

<ol> <li>All leading entries are one; 2. All entries above a leading entry are zeros (except row three).</li> </ol> Signup and view all the answers

Where do free variables come from?

<p>Columns with no pivot point.</p> Signup and view all the answers

When is a system of linear equations consistent with more than one solution?

<ol> <li>There is no pivot point in the rightmost column of the augmented matrix representative of the system; 2. If there are no free variables in a consistent system, there is a unique solution. Otherwise, the solution has infinitely many solutions.</li> </ol> Signup and view all the answers

What is a vector?

<p>A vector is a matrix consisting of a single column (and one or more rows).</p> Signup and view all the answers

For two vectors u and v to be considered equal, they must have?

<ol> <li>Same number of rows; 2. Corresponding entries must be equal.</li> </ol> Signup and view all the answers

In general, to add vectors?

<ol> <li>They must be the same size; 2. Corresponding entries are added to form a new vector.</li> </ol> Signup and view all the answers

What is scalar multiplication?

<p>Scalar multiplication uses a real number c and a vector [v1, v2, ... , vn] so that c*[v1, v2, ... , vn] = [cv1, cv2, ... , cvn].</p> Signup and view all the answers

What is denoted R to the nth power (R^n)?

<p>The set of all nx1 vectors with real number entries.</p> Signup and view all the answers

What are the 8 structural properties of R to the nth power?

<ol> <li>Vector addition is cumulative; 2. Vector addition is associative; 3. Zero vector; 4. Additive inverse; 5. Left distribution; 6. Quasi Distribution I; 7. Scalar multiplication is associative; 8. Scalar multiplication has a multiplicative identity.</li> </ol> Signup and view all the answers

Vector addition is commutative?

<p>True</p> Signup and view all the answers

Vector addition is associative?

<p>True</p> Signup and view all the answers

A set consisting of a single vector is linearly independent?

<p>False</p> Signup and view all the answers

If the columns of the nxn matrix A are linearly independent, and b⊆R^n, then the equation Ax=b has a unique solution?

<p>True</p> Signup and view all the answers

Study Notes

Linear Equations

  • A linear equation with "n" variables is expressed in the form (a_1x_1 + a_2x_2 + \ldots + a_nx_n = b), where (a_1) to (a_n) are real numbers and (b) is also real.

Solutions to Equations

  • An ordered "n-tuple" ((c_1, c_2, \ldots, c_n)) is a solution if substituting (x_i) with (c_i) satisfies the equation.

Systems of Linear Equations

  • A system consists of multiple linear equations considered together, with a solution being valid across all equations.

Theorem on Solutions

  • Systems of linear equations can have one unique solution, infinitely many solutions, or no solutions at all.

Consistent and Inconsistent Systems

  • Consistent systems yield unique or infinitely many solutions, while inconsistent systems result in no solutions.

Matrices

  • An (m \times n) matrix is a rectangular array with (m) rows and (n) columns.

Elementary Row Operations

  • Scaling: Multiply all entries of a row by a non-zero constant.
  • Swap: Exchange the position of two rows.
  • Replacement: Multiply a row by a non-zero scalar and add it to another row, replacing that row.

Row Echelon Form

  • A matrix is in row echelon form if:
    • The leading entry of each row is to the right of the leading entry in the preceding row.
    • All entries below each leading entry are zero.
    • Any rows of all zeros are at the bottom.

Reduced Row Echelon Form

  • Additional conditions for reduced row echelon form include:
    • All leading entries are 1.
    • All entries above each leading entry are zero.

Free Variables

  • Occur in columns of a matrix that do not contain pivot points.

Linear Combinations

  • A vector (b) is a linear combination of other vectors if (b = c_1v_1 + c_2v_2 + \ldots + c_pv_p) for some scalars (c_i).

Vectors

  • A vector in (R^n) is an (n \times 1) matrix, comprised of a single column.

Vector Operations

  • Two vectors are equal if they have the same number of rows and corresponding entries are equal.
  • Vector addition requires both vectors to be the same size, adding corresponding entries.

Scalar Multiplication

  • Defined for a scalar (c) and a vector (v), resulting in (c \cdot v), which scales each entry of (v).

Span of Vectors

  • The span of vectors (v_1, v_2, \ldots, v_p) in (R^n) is the collection of all possible linear combinations of these vectors.

Linear Independence

  • A set of vectors is linearly independent if the only solution to their linear combination equaling zero is the trivial solution (all coefficients are zero).
  • A set is linearly dependent if at least one vector in the set can be expressed as a linear combination of the others.

Matrix Equations

  • For an (m \times n) matrix (A) and vector (x \in R^n), the product (Ax) creates a linear combination of the columns of (A).

Linear Transformations

  • A transformation (T) mapping (R^n) to (R^m) can be defined with a matrix (A) such that (T(x) = Ax).

Properties of Linear Transformations

  • Transformation is linear if it satisfies:
    • (T(u + v) = T(u) + T(v))
    • (T(cu) = cT(u)) for any scalar (c).

Theorems on Vector Spaces

  • A linear combination (b) is valid if (b) lies within the span of the provided vectors.
  • A single vector can be attacked by the linear independence criterion; if it is non-zero, it is considered linearly independent.

Implications of Solutions

  • If Ax = b has a solution, then (b) is necessarily in the span of the columns of matrix (A.
  • If a set of vectors has more vectors than its dimensionality (more than "n" in a set of nx1 vectors), then the set is linearly dependent.

Conclusion

  • Understanding these concepts provides a foundational overview of linear algebra, crucial for solving equations, analyzing vector spaces, and applying transformations effectively.

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Prepare for your Linear Algebra midterm exam with these flashcards that cover key concepts and definitions. Understand linear equations and solutions to equations using variables. Perfect for quick revisions and enhancing your grasp of the subject matter.

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