Exam 2

Question 1

To determine if a subset S is a subspace, apply the following steps.
Step 1. Check if 0 is in S. If not, then S is not a subspace.
Step 2. If you can show that S is generated by a set of vectors, then by Theorem 4.2 S is a subspace.
Step 3. Try to verify that conditions (b) and (c) of the definition are met. If so, then S is a subspace. If you cannot show that they hold, then you are likely to uncover a counterexample showing that they do not hold, which demonstrates that S is not a subspace.

Answer 1

Subspace

Question 2

If A is an n x m matrix, then the solution to Ax=0 is called the null space of A and is denoted by null(A)

Answer 2

Null space of a matrix A

Question 3

The kernel of T is the set of vectors x such that T(x)=0

Answer 3

Kernel of a linear transformation

Question 4

A set B={u_1……u_m} is a basis for the subspace S if:

a. B spans s
b. B is linearly independent

Answer 4

Basis

Question 5

Let S be a subspace of R^n. Then the dimension of S is the number of vectors in any basis of S

Answer 5

Dimension

Question 6

The nullity of a matrix A is the dimension of the null space of A and is denoted by nullity(A)

Answer 6

Nullity of a matrix

Question 7

Let A be a nxm matrix:
a. The row space of A is the subspace Rm spanned by the row vectors of A and is denoted by row(a)

b.The column space of A is the subspace of Rn spanned by the column vectors of A and is denoted by Col(A)

Answer 7

Row space and column space a matrix

Question 8

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

Answer 8

Rank of a matrix A

Question 9

Let A be an nxm matrix. Then the nonzero vector u is an eigenvector of A if there exists a scalar(lambda) such at
Au=(lambda)u
The scalar lambda is called the eigenvalue of A

Answer 9

Eigenvalue and Eigenvector

Question 10

Let A be a square matrix with eigenvalue lambda. The subspace of all eigenvectors associated with lambda, together with the zero vector, is called the eigenspace

Answer 10

Eigenspace

Question 11

The polynomial that we get from the det(A-lambda(I))

Answer 11

Characteristic Polynomial