Linear Algebra Chapter 1 Flashcards

Questions and Answers

What is a Linear Equation?

Describes a first order line in R^n equations of the form a1x1 + a2x2 + ... + anxn = C, where a1...an are R and x1...xn are variables.

What is a System of Linear Equations?

A collection of 1 or more linear equations to be solved simultaneously.

What is a Solution in the context of linear equations?

A set of real numbers s1...sn such that each equation holds if we set x1 = s1, ..., xn = sn.

What is a Solution Set?

A set of all solutions in a system. Signup and view all the answers

What is a Consistent System?

At least one solution exists. Signup and view all the answers

What is an Inconsistent System?

No solutions exist. Signup and view all the answers

What are Equivalent Systems?

Two systems that have the same solution set. Signup and view all the answers

How do you solve a basic system?

Use interchanging, scaling, and replacement to isolate the variable and then plug into other variables. Signup and view all the answers

What does Interchange refer to in solving systems?

Reorder equations. Signup and view all the answers

What does Scaling involve?

Multiply each equation by a constant. Signup and view all the answers

What is Replacement in solving systems?

Replacing one equation with itself and a multiple of another. Signup and view all the answers

What is a Matrix?

A rectangular array of numbers with m rows and n columns. Signup and view all the answers

What is an Augmented Matrix?

A matrix that includes the set of constant solutions (numbers without variables). Signup and view all the answers

What is involved in Solving a matrix?

Try to make each pivot equal to one, with each pivot being the only number in its column. Signup and view all the answers

What is Echelon Form?

<ol> <li>All 0 rows at the bottom of the matrix. 2. Each leading nonzero entry in a row to the right of the leading nonzero entry in the previous row. 3. All entries in the column beneath a leading nonzero entry are 0's.</li> </ol> Signup and view all the answers

What is Reduced Echelon Form?

All the requirements for Echelon Form plus that each leading nonzero entry is 1 and is the only nonzero entry in its column. Signup and view all the answers

What is a Pivot in a matrix?

Leading nonzero entry in an echelon matrix. Signup and view all the answers

What is a Pivot Position?

Position of the pivot in an echelon matrix. Signup and view all the answers

What is a Basic Variable?

A variable corresponding to a column with a pivot. Signup and view all the answers

What is a Free Variable?

A variable corresponding to a column with no pivot; it can be any number. Signup and view all the answers

What makes a system solvable?

The echelon form of the matrix has no row of the form [0...0 C] where C is not equal to 0. Signup and view all the answers

What makes a solution consistent?

The solution is unique and there are no free variables. Signup and view all the answers

What is a Vector?

Quantities with direction and magnitude. Signup and view all the answers

What is a Scalar?

Quantity with only magnitude. Signup and view all the answers

What is Scalar Multiplication?

Multiplication of scalar and vector; the answer is a vector, same direction if scalar is positive, opposite if scalar is negative. Signup and view all the answers

What is a Linear Combination of a Vector?

Given v1...vk, the sum vector y = c1v1 + ... + ck vk. Signup and view all the answers

What are the weights in a Scalar Combination of a Vector?

'Weights' of linear combination. Signup and view all the answers

What is a Span in linear algebra?

Given a collection of vectors {vi...vk} in Rn, the collection of vectors in Rn which can be expressed as y = c1v1 + ... + ck vk. Signup and view all the answers

What is Matrix Multiplication?

Let A be an m x n matrix with columns a1...an. Let x be a vector which is the linear combination of columns a, with weights Ax = x1a1 + x2a2 + ... + anxn. Signup and view all the answers

What is the formula for matrix multiplication?

Ax = b where A is the m x n coefficient matrix, x is a variable vector, and b is the vector of constant terms. Signup and view all the answers

What are the equivalent statements of a m x n matrix where the vector x is real?

<ol> <li>For all real vectors b, Ax = b is solvable. 2. Every real b is a linear combination of columns of A. 3. The columns of A span R^m. 4. Every row of A has a pivot.</li> </ol> Signup and view all the answers

What does Unique mean in the context of linear equations?

Pivot in every column means that all variables are basic. Signup and view all the answers

What is a Homogeneous System?

Ax = 0 where 0 is the zero vector; always solvable for any matrix A, just make x = 0. Signup and view all the answers

What is a Nontrivial solution?

A solution to a system that makes the system equal to the zero vector, but is not the zero vector itself. Signup and view all the answers

What is a Trivial solution?

A solution that can only equal 0 with the 0 solution. Signup and view all the answers

What is Linear Independence?

A collection of vectors {v1..vk} in some Rn whose linear combination has only the trivial solution. Signup and view all the answers

What is Linear Dependence?

A collection of vectors {v1..vk} in some Rn whose linear combination has more than just the trivial solution. Signup and view all the answers

What are some facts about Linear Dependence?

<ol> <li>The 0 vector is dependent. 2. If the 0 vector is an element of {v1...vk}, dependent. 3. If vectors are multiples, dependent. 4. If {v1..vk} in Rn and k > n, dependent.</li> </ol> Signup and view all the answers

What is a Linear Transformation?

T: Rn -> Rm is a function taking vectors in Rn to vectors in Rm. Signup and view all the answers

What is the Domain of Transformation?

T: Rn -> Rm is Rn. Signup and view all the answers

What is the Co-Domain of Transformation?

T: Rn -> Rm is Rm. Signup and view all the answers

What is the Range of Transformation?

Subset of vectors in Rm which are T(x) for some vector x in Rn. Signup and view all the answers

What makes a transformation linear?

<ol> <li>T(u + v) = T(u) + T(v). 2. T(cu) = cT(u).</li> </ol> Signup and view all the answers

What does Onto mean in the context of transformations?

A T: Rn -> Rm whose image is all of Rm; recall b in Rm so that there is x in Rn with T(x) = b. Signup and view all the answers

What does One-to-One mean in transformations?

A T: Rn -> Rm which if T(x1) = T(x2) implies x1 = x2. Signup and view all the answers

What is Matrix Multiplication?

Let A be an m x n matrix and B be an n x k matrix. Then the product is given by AB = matrix Ab1 where b1 is a column of B. Signup and view all the answers

What are the Properties of Matrix Multiplication?

<ol> <li>A(BC) = AB(C). 2. A(B + C) = AB + AC. 3. r(AB) = (rA)B = A(rB). 4. If A (mxn) and ImA = A = AIn.</li> </ol> Signup and view all the answers

What is a Power Matrix?

Multiply by itself; can only happen with a square matrix. Signup and view all the answers

What is Transposition?

Flips matrix; rows become columns and columns become rows, denoted At. Signup and view all the answers

What are the Properties of Transposition?

<ol> <li>(At)t = A. 2. (A + B)t = At + Bt. 3. (rA)t = r(At). 4. (AB)t = BtAt; must swap order!</li> </ol> Signup and view all the answers

What is an Invertible Matrix?

A is invertible if there is some nxn matrix B such that AB = In = BA; we denote B by A^-1. Signup and view all the answers

What is a Singular Matrix?

A matrix that is not invertible. Signup and view all the answers

What makes a matrix invertible?

If A = [a b; c d] is a 2x2 matrix then A is invertible iff ad - bc is not equal to 0, then A^-1 = 1/(ad - bc) [d -b; -c a]. Signup and view all the answers

What are the Properties of an Invertible Matrix?

<ol> <li>ax = b has one solution for every b in Rn. 2. Every matrix has n pivot positions. 3. If A is invertible, so is A^-1. 4. (AB)^-1 = B^-1A^-1. 5. If A is invertible, then so is At.</li> </ol> Signup and view all the answers

How do you find an inverse?

[A In] -> [In A]; the last three columns are the inverse. Signup and view all the answers

What is an Elementary Matrix?

A matrix that is obtained from the identity matrix with a single row operation. Signup and view all the answers

What are determinants?

Of [a] is a; A = a11detA11 - a12detA12 + ... + (-1)^n a1n detA1n. Signup and view all the answers

For what types of matrices are determinants defined?

Square matrices. Signup and view all the answers

What does it mean if the determinant is non-zero?

The matrix has an inverse. Signup and view all the answers

What if the matrix has a 0 row or 0 column?

The determinant is 0. Signup and view all the answers

What is the Cofactor of A?

The ijth cofactor of A is (-1)^(i+j) Aij. Signup and view all the answers

What is Cofactor Expansion?

The process of finding a determinant. Signup and view all the answers

What is an Upper Triangular Matrix?

|x1 # # | 0 |x2 # | 0 0 x3 ... Signup and view all the answers

What is a Lower Triangular Matrix?

|x1 0 0 | # x2 0 | # # x3 ... Signup and view all the answers

How do you find the determinant of a triangular matrix?

Multiply the diagonal leading entries. Signup and view all the answers

What are the effects of each row operation on the determinant?

Replacement = nothing; exchange = multiply by -1; scaled by k = multiply by k. Signup and view all the answers

How do you compute the determinant with a triangular matrix?

det A = (-1)^r * product of pivots, where r is the number of pivots. Signup and view all the answers

What are three random facts about determinants?

<ol> <li>det At = det A. 2. det(AB) = detA * detB for A and B that are nxn matrices. 3. det(A^-1) = 1/detA.</li> </ol> Signup and view all the answers

Study Notes

Linear Equations and Systems

Linear equations represent first-order lines in R^n, following the form a1x1 + a2x2 + ... + anxn = C, where a1...an are real numbers and x1...xn are variables.
A system of linear equations consists of two or more linear equations that need to be solved simultaneously.
A solution is a set of real numbers s1, s2, ..., sn that satisfies all equations when substituted for the variables.
The solution set includes all possible solutions to a system of equations.

System Consistency

A consistent system has at least one solution.
An inconsistent system has no solutions.
Equivalent systems share the same solution set.

Solving Systems

Basic methods include interchanging, scaling, and replacement to isolate variables.
Interchanging refers to reordering equations; scaling involves multiplying equations by constants; replacement replaces an equation with a multiple of another.

Matrices

A matrix (m x n) is a rectangular array of numbers with m rows and n columns.
An augmented matrix combines the coefficient matrix with the constant solutions.
To solve a matrix, pivot positions are established; each pivot should equal one, with no other entry in its column.

Matrix Forms

Echelon Form has all zero rows at the bottom and leading nonzero entries arranged progressively rightwards, with zeroes below each leading entry.
Reduced Echelon Form maintains the properties of Echelon Form plus requiring each leading entry to be 1 and the only nonzero entry in its column.

Variables and Pivots

A pivot is the leading nonzero entry in an echelon matrix, while its position is called the pivot position.
Basic variables correspond to columns with pivots; free variables correspond to columns without pivots and can take any value.

Solution Criteria

A system is solvable if its echelon form does not contain a row in the format [0...0 C] where C is nonzero.
A unique solution occurs when there are no free variables and each variable is basic.

Vectors and Scalars

A vector consists of quantities with direction and magnitude, while a scalar contains only magnitude.
Scalar multiplication combines a vector with a scalar, affecting the vector's direction based on the scalar's sign.

Span and Transformations

The span of vectors {v1...vk} is the collection of all possible linear combinations of these vectors.
Linear transformations, T: R^n -> R^m, map vectors from one space to another.

Properties of Transformations

A linear transformation satisfies the following: T(u + v) = T(u) + T(v) and T(cu) = cT(u).
A transformation is onto if its image covers all of R^m; it's one-to-one if distinct inputs yield distinct outputs.

Matrix Operations

Matrix multiplication involves combining mxn matrix A with nxk matrix B to form a product matrix.
Properties of matrix multiplication include associativity and distributive properties, while it is not commutative.

Inverses and Determinants

An invertible matrix A has a corresponding inverse B such that AB = BA = In (identity matrix).
Determinants are defined for square matrices, indicating whether a matrix is invertible (nonzero determinant) or singular (zero determinant).

Special Matrix Types

Triangular matrices are categorized into upper and lower triangular forms.
The determinant of a triangular matrix equals the product of its diagonal elements.

Cofactor and Expansion

The cofactor of a matrix A is defined as (-1)^(i+j) Aij.
Cofactor expansion applies to calculate determinants.

Properties of Determinants

Determinant properties include det(AB) = det(A)det(B) and det(A^T) = det(A).
Row operations affect determinants: replacement has no effect, while row exchanges flip the sign and scalar multiplication modifies the determinant by that scalar factor.

General Facts

The zero vector is considered dependent, and if the collection includes the zero vector or k > n, it indicates linear dependence.
Linear independence of a vector collection exists if only the trivial solution can be formed from their combination.

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Test your knowledge of key concepts in Linear Algebra with these flashcards from Chapter 1. Each card defines important terms like Linear Equations and Systems of Linear Equations that are fundamental to the subject. Perfect for quick revision and learning!