Linear Algebra Flashcards
A subspace of Rn is any collection S of vectors in Rn such that:
(i) The zero vector 0 is in S.
(ii) If u and v are in S, then u + v are in S. (S is closed under addition.)
(iii) If u is in S and c is a scalar, then cu is in S. (S is closed under scalar multiplication
A basis for a subspace S of Rn is a set of vectors in Rn that:
(i) spans S and
(ii) is linearly independent.
Let A be an m × n matrix. The null space of A is…
The subspace of Rn consists of
solutions to the homogeneous linear system Ax = 0
True or False.
Using that 1/|z| is a positive real number for z ̸= 0.
True. Explain
True or False.
Matrix Multiplication is a row by column operation.
True. Explain
True or False
If A is a square matrix, then A 1 AT is a symmetric matrix.
True. Explain
True or False.
For any matrix A, AAT and ATA are symmetric matrices.
True. Explain
If A is an invertible n x n matrix, then the system of linear equations given by Ax = b has the unique solution x = A-1b for any b in Rn.
True. Explain
Let A be a matrix whose entries are real numbers. For any system of linear equations Ax b, exactly one of the following is true:
a. There is no solution.
b. There is a unique solution.
c. There are infinitely many solutions.
True. Explain
A transformation T: Rn S Rm is called a linear transformation if
1. T(u + v) =T(u) + T(v) for all u and v in Rn and
2. T(cv) = cT(v) for all v in Rn and all scalars c.
True
Elementary Row Operations has 3 steps name them.
1) Interchange Rows
2) Mutiltply a row
3) Add a multiple of a row to another
True or False.
A system of linear equations with augmented matrix [A|B] is consistent if and only if b is a linear combination
True
True or False.
A square matrix is symmetrical if AT = A that is if A is equal to its own transpose
True
How do you find the inverse of a 2 x 2 matrix and verify it? List the steps?
1) Find the determinant
2) Multiply the inverse by the determinant
3) Verify by multiplying the original matrix A by the newly found inverse.
Solving systems using the inverse.
X = A-1b. Name each part of the proof.
X = The coeffecients
A-1 = the inverse
b = Solutions of the systems
True or False.
If A is an invertible matrix so is the AT and (AT)-1 = (A-1)T
True
Steps to find the corresponding eigenvectors with a given eigenvalue.
1) Find the null space of matrix A [A - eigenvalue(identity)
2) Row reduce
3) Get leading 1’s
4) Use the free variable make equal to t or s
5) Find the span
Determine the eigenvalues of following matrix. Analyze the steps.
1) Det(A - lambda(I)) = Multiply each diagonal by (-lambda)
2) Use cofactor expansion
3) Calculate each determinate
4) Simplify and solve for each eigenvalue
Determine the eigenvectors from the eigenvalues found. State the steps.
1) Row reduce and find the leading 1’s
2) Then find the span
True or False.
Every 2 x 2 with eigenvalues 0 and 1 is diagonalizable.
True. If the 2 x 2 matrix has 2 distinct eigenvalues, then it is diagonalizable
True or False.
Every 3 x 3 martix with eigenvalues 0 and 1 is invertible.
False. It has an eigenvalue an eigenvalue of 0
True or False. If you have an eigenvalue of 0 then your matrix is invertible.
False. If you have an eigenvalue of 0 then your 3 x 3 matrix is not invertible.
True or False.
If a is a square matrix and det(A) /= ), then the rows of A are linearly independent
True.
True or False.
Every upper-triangle square matrix is diagonalizable.
False.
True or False.
Every upper triangular matrix does not have any complex eigenvalues.
True.
How do you know that the det = 0 of a matrix A without finding the actual determinant?
1) If a matrix is linearly independent is non zero
2) If a matrix is linearly dependent than the det = 0
True or False.
The columns of a m x n matrix form an orthonormal set if and only if QTQ = I
True.
A square matrix Q is orthogonal if and only if Q-1 = QT
True
What are we doing when we normalize orthogonal vectors?
We are turning them into unit vectors.
True or False.
If A is a symmetric matrix than the eigenvalues of a are real.
True.
True or False.
For any n x n matrix A, A and the rref(A) will have the same determinant
False.
True or False.
For any n x n matrix A, A nd rref(A) will have the same eigenvectors
False
