Linear Algebra Final Exam Review

1\.1 When are two systems equivalent?

When they have the same solution set.

1\.1 When is a system consistent?

When it has exactly one solution or infinitely many solutions.

1\.1 When is a system inconsistent?

When there is no solution.

1\.2 When do you use Reduced Echelon Form instead of Echelon Form?

When you want a solution.

1\.2 What is a pivot position?

A location in A that corresponds to a leading 1 in the reduced echelon form of A.

1\.2 What is a pivot column?

A column of A that contains a pivot position.

1\.2 What is a free variable?

The variable that does not correspond to a pivot column and leads to infinitely many solutions.

1\.3 What is a column vector or vector?

A matrix with only one column.

1\.4 When does the matrix equation Ax=b have a solution?

If and only if b is a linear combination of the columns of A.

1\.4 What are the four ways of writing a linear system?

Matrix equation Ax=b, Vector equation x1\[\]+x2\[\].., Augmented matrix \[a1 a2 b\], System of equations x1 + x2 = b

1\.5 What is the trivial solution?

x=0, the zero vector.

1\.5 When does the homogeneous equation Ax=0 have a nontrivial solution?

If and only if the equation has at least one free variable.

1\.5 What is parametric vector form?

Writing the solutions to Ax=0 as the free variable times the solutions written as a vector, where the free variable is represented as a 1. Can also be used for a nonhomogeneous linear system that has many solutions/free variables.

1\.7 When is a set of vectors said to be linearly independent?

When the homogeneous vector equation has only the trivial solution/no free variable.

1\.7 When is a set of vectors said to be linearly dependent?

When there is at least one free variable.

1\.7 How to tell if a set of vectors are linearly dependent by observation.

1) A set of two vectors is linearly dependent if and only if one vector is a multiple of the other.

2) If a set has more vectors than there are entries in each vector then the set is linearly dependent.

3) If a set contains the zero vector, then the set is linearly dependent.

1\.8 How do you find the image of u under the transformation of T?

You multiply the columns of the matrix A by the entries of the vector u and add them to find T(u).

1\.8 How do you find an x whose image under T is b and determine if x is unique?

Create an augmented matrix Ax=b, reduced echelon form, and the vector solution is the answer. If there is no free variable, x is unique.\*

1\.8 How do you determine if b is in the range of the linear transformation x →Ax?

Create augmented matric and get into echelon form. If b is a solution of Ax then b is in the range of the transformation.

1\.9 How do you find the standard matrix of T?

Write the ordered pairs of T(en) as columns of A or write the coefficients of the given system as the rows of A.

1\.9 When is T onto?

When there is a pivot in every row.

1\.9 When is T one-to-one?

When the columns are linearly independent, have no free variables.

2\.1 When do A and B commute with one another?

When AB=BA.

2\.2 Invertible matrix theorem shortcut

1/(ad-ba)\[d-b,-c a\]

2\.2 When is A not invertible?

If ad-bc=0

2\.3 What are the parts of the invertible matrix theorem?

A is an invertible matrix.

A is row equivalent to the identity matrix.

A has a pivot in every row.

Ax=0 has only the trivial solution.

The columns of A form a linearly independent set.

x→Ax is one-to-one.

The columns of A span R^n.

x→Ax maps R^n onto R^n.

AT is an invertible matrix.

2\.3 Can a square matrix with two identical columns be invertible? Why or why not?

No, because the columns are linearly dependent.

3\.1 In cofactor expansion, when do you use a positive first coefficient?

When using an odd row or column.

3\.1 In cofactor expansion, when do you use a negative first coefficient?

When using an even row or column.

3\.2 How does replacement effect the determinant?

det(A)=det(B)

3\.2 How does interchanging two rows of A to produce B change the determinant?

detB=-detA and detA=-detB

3\.2 How does scaling A to produce B effect the determinant?

detA= 1/kdetB

3\.3 What is Cramer’s rule?

Replace the columns of A with b one at a time and find the determinant of each one, then divide by the determinant of A to find each x.

2\.8 What forms the basis for Col(A)?

The pivot columns of matrix A.2

2\.8 What forms the basis for Nul(A)?

The parametric vector form of the solution of Ax=0.

2\.8 What is the dim(Nul(A))?

The number of free variables in Ax=0.

2\.8 What is the dim(Col(A))?

The number of pivot columns in A.

2\.8 What is the rank(A)?

rank(A)=dim(Col(A))

2\.9 Rest of invertible matrix theorem.

The columns of A form a basis of R^n.

Col(A)=R^n.

rank(A)=n

dim(Nul((A))=0

Nul(A)={0}

2\.9 How to find the dimension of solution space of Ax=0 given the rank?

rank(A) + dim(Nul(A)) = n

2\.9 If A is one-to-one, what is dim(Nul(A))?

0

5\.1 Invertible matrix theorem continued.

The number 0 is not an eigenvalue of A.

The determinant of A is not equal to 0. 5

5\.1 What are the eigenvalues of a triangular matrix?

The entries on its main diagonal.

5\.3 How do you form the matrix D?

The eigenvalues on the main diagonal in the order they appear in P and zeros elsewhere.

5\.3 How is P formed?

By the eigenvectors of A.

5\.5 How do you form the matrix P with complex eigenvalues?

\[Real part, Imaginary part\]

5\.5 How do you form C with complex eigenvalues?

a +- bi, C=\[a -b, b a\]

5\.6 How do you find the solution of xk+1=Axk?

x0=c1v1+c2v2… create augmented matrix with v’s and x0, REF to find c’s. xk=c1(lambda1)^kv1….

5\.7 What is x(t) equal to?

c1v1e^lambda1t…

6\.1 How do you find the length of a vector?

Take the square root of v dot v. If distance between u and v, it is the length of u-v.

6\.1 When is a pair of vectors orthogonal?

When their inner product is equal to zero.

6\.1 When is a set orthonormal?

When it is an orthogonal set of unit vectors (the length of each vector is 1).

6\.1 How do you make an orthogonal set orthonormal?

Divide each entry in the vectors by its length.

6\.1 How do you find the orthogonal projection of y onto the line through u and the origin?

Find y^= (y dot u)/(u dot u)u then find z = y - y^, then write y = y^ + z

6\.1 How do you find the distance form v to span{w}?

v^=(v dot w)/(w dot w)w then find the distance of v-v^.

6\.3 What are three ways to describe y^?

The orthogonal projection of y onto W.

The closest point to y in W.

The best approximation to y by elements of W.

6\.3 How to work: Write v as the sum of two vectors, one in Span{u1} and the other in span{u2 u3 u4}.

Span{u1}=(v dot u1)/(u1 dot u1)u1.

v2^=v-v1^

v= v1^+v2^

6\.3 How to solve: Find the orthogonal projection of y onto Span{u1 u2}.

y^= partial grandschmit u1 + u2

6\.3 How to solve: Find the closest point to y in the subspace W spanned by v1 and v2 then find the distance from y to W.

y^= partial grandschmit v1 + v2

Square root of y-y^

6\.3 How to solve: Find the distance from y to the plane in R^3 spanned by u1 and u2.

y^= partial grandschmit u1 + u2

Square root of y-y^

6\.4 How to solve: Find an orthonormal basis for a subspace whose basis is given.

v1=x1.

v2= x2-(x2dotv1)/(v1dotv1)v1

Square root of each vector, divide each entry in v1 and v2 by that vectors length.

6\.4 How to solve: Find an orthonormal basis for the column space of the following matrix A.

Grand schmit process for all three vectors, then find length of each vector to create the basis.

6\.4 How is Q found for QR factorization?

The orthonormal vectors resulting from the grandschmit process.

6\.4 How is R found for QR factorization?

QTA

6\.5 What is the equation for least squares solutions?

x^=(ATA)^-1(ATb)

6\.5 What is the equation for least squares error?

Square root of b-Ax^

6\.5 How to solve: Describe all least squares solutions of Ax=b.

Find ATA and ATb. Create the augmented matrix Ax=b and REF. Write solution in parametric vector form.

6\.5 How to solve: Find the orthogonal projection of b onto Col(A) and a least squares solution of Ax=b.

Find b^=(b dot a)/(a dot a)a for each column in A.

To find the least squares solution, find which values of x^ would make Ax=b. The values are the coefficients found from (b dot a)/(a dot a) for each column.

6\.5 What equation do you use to find the least squares solution of Ax=b using QR factorization?

x^=R^-1QTb

6\.6 How to solve: Find the equation y=B0+B1x of the least squares line that best fits the given data points.

B^=(XTX)^1XTy^.

X\[1s, x values\]

y^=\[y values\]

B^=\[B1, B2\]

6\.6 How to solve: Find fit of y=B1x+B2x^2

X=\[x values, x values squared\]

y^=\[y values\]

B^=(XTX)^1XTy^.

B^=\[B1, B2\].