Linear Algebra Final Review

Suppose A is an m x n matrix.
How do we determine the number of pivotal columns?

The number of pivotal columns are determined by the number of columns with non-zero leading entries(after you reduce).

Suppose A is an m x n matrix.
What do the pivotal columns tell us about the solution to the equation Ax=b.

That there is either a unique solution, no solution or infinite solution.

Suppose A is an m x n matrix.
What space is equal to the span of the pivotal columns.

The null space and col space are equal to the span of the pivotal columns.

Suppose A is an m x n matrix.
Is it possible to write A = LU, where L is a lower triangular matrix and U is an upper triangular matrix? If not, what additional information do you need?

No, we need to know if A is a square matrix and if the determinant does not = 0.

Suppose A is an m x n matrix.

Is it possible to write A = QR, where Q has orthonormal columns and R is an upper triangular matrix? If not, what additional info do you need?

Yes, all we need is for Q to be orthonormal and R to be upper triangular. It doesnt matter if A is square or not.

Suppose A is an m x n matrix.
What is the difference for solving Ax = b and Ax=0? How are these two solutions related graphically?

Ax=b solves for the unique solution(non-homogenous) and Ax=0 solves for the trivial solution(homogenous). The equations result in two lines that are parallel but are offset by a vector.

Suppose A is an m x n matrix.
If rank(A) = r, where 0 < r <= n, how many columns are pivotal. What is the dimension of the solution of the space to Ax = 0.

r columns are pivotal and the solution to the null space of A is n-r.

Suppose that T is a linear transformation T : R^n —> R^m with the associated matrix A.
What are the dimensions of A?

mxn

Suppose that T is a linear transformation T : R^n —> R^m with the associated matrix A.

How do we find TA(x)?

A = [T(e1) T(e2)..T(e3)]

Suppose that T is a linear transformation T : R^n —> R^m with the associated matrix A.

Using the matrix A, How would we know if TA is one to one?

If the columns of A are linearly independent

Suppose that T is a linear transformation T : R^n —> R^m with the associated matrix A.
How do we find the range of Ta

The range of T_a is the col space of A

Suppose A is an n x n invertible matrix

What can you say about the columns of A?

The columns of A are linearly independent and span R^n.

Suppose A is an n x n invertible matrix

What is rank(A) and nullity(A)

rank(A) = n, nullity(A) = 0

Suppose A is an n x n invertible matrix

what do you know about the det(A)

The det of A is non Zero

Suppose A is an n x n invertible matrix

How many solutions are there to Ax = b?

There is exactly one solution in R^n.

Suppose A is an n x n invertible matrix

What is the nullspace of A?

The subspace of R^n, the set of homogenous solutions to Ax = 0, meaning it contains only the zero vector.

Suppose A is an n x n invertible matrix

Do you know anything about the eigenvalues of A?

None of the eigen values can be 0.

Suppose A is an n x n invertible matrix

Do you know weather or not A is diagonalizable?

It is diagonalizable if there are n distinct eigenvalues and n eigenvectors.

Suppose A is an n x n invertible matrix
How can we use the columns of A to create an orthogonal basis for R^n?

We can use the Gram Schmidt process to find an orthogonal basis for the R^n.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)
What is the degree of p(λ)?

The degree would be n since A is an n x n matrix.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

Counting multiplicities, how many eigen values will A have?

n eigenvalues

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

If p(0) = 0, what do you know about the matrix A?

This equation means the determinant is 0. Which means that matrix A is linearly independent and not invertible.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

How will you know if A is diagonalizable?

If there are n distinct eigenvalues.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

If A is a 3 × 3 matrix and has complex eigen values, how many real roots are there to p(λ)?

Since its a 3×3 matrix, there would be 3 roots.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

Suppose p(C) = 0 for some real number C. How do you find the values of x for which Ax = Cx?

You need to find the eigenvectors of A which would be equivalent to C.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

How do you find the sum and product of the eigen values?

To find the product you need to find the determinant of the matrix A. To find the sum you need to find the trace of matrix A which is the sum of the diagonal entries.

Suppose A is an n x n matrix with characteristic equation p(λ) = det(A - λI)

If A is NOT triangular or diagonal, do the solutions of p(λ) change when A is reduced to echelon form?

No, it doesn’t change. You would still have the same determinant after reducing to echelon form.

If two matrices A, B satisfy AB = BA and A is invertible, then A^-1B = BA^-1

True. Since AB = BA, if you multiply both sides by A^-1 the equality is still true. Which means that AB = BA and A is invertible, then A^-1B = BA^-1 is true.

Any plane in R³ is a subspace.

False, for a plane to be a subspace of R³ it must pass through the origin(zero vector) which is one of the necessary conditions.

If A matrix is invertible and det A = 3, the det A^-1 is 1/3.

True. The det(A^-1) = 1/det(A) if and only if matrix A is invertible.

If the rank of a n x n matrix is n, then the columns of A form a basis of Rⁿ

True. If the matrix A is invertible and its rank is n. That means the col space of A spans the entire space of Rⁿ and forms a basis of Rⁿ

If {v₁,v₂} is linearly independent and {v₁,v₂, v₃} is linearly dependent, then v3 is in the span of {v₁,v₂}

True. This statement means that v3 can be written as a linear combination of v1 and v2 which would mean it lies in the span of {v1,v2}.

The columns of A = [1 -3 -4; -3 -2 6; 5 -1 -8] span R³

False. This is because the determinant of A is zero and there are no pivots in each column which is the conditions to span R³

The columns of matrix A are linearly independent if Ax = 0 has the trivial solution.

False. For A to be linearly independent, Ax = 0 has to have only the trivial solution.

The columns of a 5 × 6 matrix A are linearly dependent.

True. Since there are more columns then rows, it is linearly dependent.

If A and B are matrices that satisfy AB = I, then both A and B are invertible.

False. For us to know that A and B are 100% invertible, we must know that BA = I which is not stated. Therefore A and B can’t be invertible.

A translation defined by T(x1, x2) = (x1 + h, x2 + h), h = non zero is a linear transformation.

False. For a translation to be a linear transformation two conditions need to be met.

T(U+V) = T(U) + T(V)
T(cV) = cT(V)

These conditions are not met.

If A is invertible, then the columns of A^-1 are linearly independent.

True. If A is invertible, that means that its columns are linearly independent. This means the same thing is true for the inverse of A.

If(B-C)D = 0, where B and C are m x n matrices and D is invertible, then B = C.

True. If we multiply both sides of the equation by D^-1 we get B-C = 0. Which gives us that B = C.

if span {v1, v2, v3} = R³ then {v1, v2, v3} is a basis for R³

True. A basis for R³ would require 3 vectors that span the space of R³. Since {v1, v2, v3} consists of 3 vectors and span the space of R³ it qualifies as a basis for R³.

The column space of a 5×7 matrix is a subspace of R⁷

False. The column space of a 5×7 matrix would only be a subspace for R⁵ since the column vectors are 5 dimensional(5 rows).

If the rank of a 3×3 matrix A is 1, then NulA must be a plane.

True. Since the rank is 1, that means the nul space of A is 2. This means the null space is two dimensional which would be a plane.

If two eigenvectors are linearly independent, then they respond to distinct eigenvalues.

False. Two eigenvectors can be linearly independent even if they correspond to the same eigenvalue

Let A and B be n x n matrices. If A is similar to B, then they have the same eigenvectors.

False. They can have the same eigenvalues but may not have the same eigenvectors. Similar matrices can have different eigenvectors

There exists a 3×3 matrix that has no eigenvectors in R³

False. Every 3×3 matrix has at least one eigenvalue and corresponding eigenvector in R³

If matrix a is diagonalizable, then det A does not = 0.

False. For a matrix to be diagonalizable, it has to be a square matrix with n distinct eigen values. The determinant is irrelevant.

Let A be a 2×2 matrix, then A can have one real and one complex eigen value.

False. In a 2×2 matrix, the pair of eigenvalues must either both be real or both be complex. You cant have both.