Linear Algebra

Question 1

5.1: Distance Between Two Vectors

Answer 1

Question 4

5.1: The Cauchy-Schwarz Inequality

Answer 4

Question 6

5.1: The Triangle Inequality

Answer 6

Question 7

5.1 The Pythagorean Thereom

Answer 7

Question 8

5.2: Definition of Inner Product

Answer 8

Question 9

5.2 Orthogonal Projection and Distance

Answer 9

Question 10

5.3 Orthonormal Basis

Answer 10

A set of vectors that are both mutually orthogonal and unit vectors.

Question 11

5.3 Gram-Schmidt Orthonormalization Process

Answer 11

1. B={v₁, v₂…,v}, a set of vectors that are the basis for an inner product space V.
2. B’={w₁, w₂…,w}, w₁=v₁, w₂=v₂-proj_v₂w, w₃=v₃-proj_v₃w₁-proj_v₃w₂; orthogonalization
3. Find the unit vectors for each w vector.

Question 13

3.1 Minor

Answer 13

Question 14

3.1 Cofactor

Answer 14

Question 15

3.1 Determinant of a Triangular Matrix

Answer 15

The product of all the entries on the principal diagonal.

Question 16

3.2 Elementary Row Operations and Determinants

Question 17

3.3 Determinant of a Matrix Product
Determinant of a Scalar Multiple of a Matrix
Determinant of an inverse Matrix
Determinant of a Transpose of a Matrix

Question 18

3.4 Adjoint of a Matrix

Answer 18

Adj(A)=the transpose of a cofactor matrix.

Question 19

3.4 Inverse of a nxn Matrix Using its Adjoint

Answer 19

Question 20

3.4 Cramer’s Rule

Answer 20

Question 21

3.4 Area of a Triangle with vertices

(x₁, y₁), (x₂, y₂), and (x₃, y₃)

Answer 21

Question 22

3.4 Two-Point Form of the Equation of a Line (x₁, y₁), (x₂, y₂)

Answer 22

Question 23

3.4 Volume of a Tetrahedron with vertices

(x₁, y₁,z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₄, y₄, z₄)

Answer 23

Question 24

3.4 Three-Point Form of the Equation of a Plane

(x₁, y₁,z₁), (x₂, y₂, z₂), and (x₃, y₃, z₃)

Answer 24

Question 25

2.3 Inverse of a 2x2 matrix

Answer 25

Question 26

2.3 Inverse of an nxn matrix

Answer 26

Question 27

2.3: Solve a system of equations

Answer 27

Question 28

2.4 Definition of an Elementary Matrix

Answer 28

A nxn matrix that can be obtained from the identity matrix I by a single elementary row operation.

Question 29

2.5 Stochastic Matrics

Answer 29

P=Probability Matrix (columns add up to 1): PX

Question 30

2.5 Leontief Input-Output Models

Answer 30

Question 31

2.5 Matrix Form for Linear Regression

Answer 31

Question 32

2.5 Encryption

Answer 32

Question 33

1.1: Row Echelon Form

Answer 33

A stair step pattern with leading coefficients of 1.

Question 34

1.2: Reduced Row Echelon Form

Answer 34

Every column that has a leading 1 has zeros in
every position above and below its leading 1.

Question 35

1.2: Gaussian Eliminination with Back Substitution

Answer 35

1. Write the augmented matrix of the system of linear equations.
2. Use elementary row operations to rewrite the matrix in row-echelon form.
3. Use back-substitution to find the solution.

Question 36

1.2: Gauss-Jordan Elimination

Answer 36

Same as Gaussian Elimination but instead of row-echelon form you rewrite the matrix in reduced row-echelon form.

Question 37

1.2: Homogeneous System of Linear Equations

Answer 37

• Systems of linear equations in which each of the constant terms is zero.
• A trivial solution is where all variables equal 0.
• Must have at least one solution, which is the trivial solution.
• If the system has fewer equations than variables, then it must have infinitely many solutions.

Question 38

1.3.: Polynomial Curve Fitting

Answer 38

Subsitute each of the given points into the polynomial function then solve for each variable.

Question 39

4.1: Properties of Vector Addition and Scalar Multiplication in Rⁿ

Answer 39

Question 40

4.1: Properties of Additive Identity and Additive Inverse

Answer 40

Question 41

4.2: Properties of Scalar Multiplication

Answer 41

Question 42

4.3: Test for a Subspace

Answer 42

Subspaces must contain the zero vector.

Question 43

4.4: Finding a Linear Combination

Answer 43

Question 44

4.5: Definition of Basis

Answer 44

A set of vectors S in a vector space V that span V and are linearly independent.

Question 45

4.5: Characteristics of Bases

1. Uniqueness of Basis Representation
2. Bases and Linear Dependence
3. Number of Vectors in a Basis

Answer 45

1. If S is a basis for a vector space V then every vector in V can be written in one and only one way as a linear combination of vectors in S.
2. If S is a basis for a vector space V then every set containing more than n vectors in V is linearly dependent.
3. If a vector space V has one basis with n vectors, then every basis for V has n vectors.

Question 46

4.5: Number of Dimensions in a Vector Space

Rⁿ, P_n, M_m,n

Answer 46

Rⁿ: n, Pⁿ:n+1, M_m,n: mn

Question 47

4.6 Basis for the Row Space of a Matrix

Answer 47

If a matrix A is row-equivalent to a matrix B in row-echelon form, then the nonzero row vectors of B form a basis for the row space of A.

Question 48

4.6: Basis for a Column Space of a Matrix

Answer 48

Nonzero row vectors of B form a basis for the row space of A^T.

Question 49

4.6: Rank of a Matrix

Answer 49

The dimension of the row (or column) space of a matrix A is called the rank of A and is denoted by rank(A).

Question 50

4.6: Nullspace

Answer 50

If A is an m x n matrix, then the set of all solutions of the homogeneous system of linear equations Ax= 0 is a subspace of Rⁿ called the nullspace of A and is denoted by N(A).

Question 51

4.6: Finding the Nullspace

Answer 51

1. Write the coefficient matrix in row-echelon form.
2. Solve for the variables, making use of parametric form.
3. Write the system as linear combinations of the variables.

Question 52

4.6: Dimension of the Solution Space

Answer 52

Rank (A)–nullity

Question 53

4.6: Finding the Solution Set of a Nonhomogenuous System of Linear Equations Ax=b

Answer 53

1. Write the augmented matrix in row-echelon form.
2. Solve for the variables, making use of parametric form.
3. Write the vectors as a linear combinations and remove the particular solution x_<em>p</em>.

Question 54

4.6: Summary of Equivalent Conditions for Square Matrices

Answer 54

Question 55

4.7: Finding a Coordinate Matrix Relative to a Standard Basis

Answer 55

Question 56

4.7: Finding a Coordinate Matrix Relative to a Nonstandard Basis.

Answer 56

1. Write x as a linear combination of the nonstandard basis u. x=c₁u₁+c₂u₂+c₃u₃.
2. Write as a system of linear equations and matrix equation.
3. Solve for the variable

Question 57

4.7: Change of Basis of Rⁿ

Answer 57

P [x]_{<em>B’ </em>}= [x]_{<em>B ; </em>}P^–1 [x]_{<em>B </em>}= [x]_{<em>B’</em>}

P is the transitional matrix

P^–1 is the inverted transitional matrix

[x]_<em>B</em> is the coordinate matrix of x relative to B

[x]_{<em>B’ </em>}is the coordinate matrix of x relative to B’

Question 58

4.8: The Wronskian

Answer 58