Linear Independence, Bases and Dimension of Subspace

Question 1

What are the three properties a subset W must have in order to be considered a (linear) subspace of R^n?

Answer 1

1. W contains the zero vector in R^n
2. W is closed under addition: if w1 and w2 are both in W, then so is w1+w2.
3. W is closed under scalar multiplication: if w is in W and k is an arbitrary scalar, then kw is in W

Question 2

What is a redundant vector?

Answer 2

We say a vector vi in the list v is redundant if vi is a linear combination of the list of vectors

Question 3

What makes a vector linearly independent?

Answer 3

Vectors are considered linearly independent if none of them is redundant. Otherwise, the vectors are considered linearly dependent.

Question 4

What makes a given number of vectors in a subspance V of R^n a basis of V?

Answer 4

Vectors v1……vm form a basis of V is they span V and are linearly independent

Question 5

How does a vector or list of vectors span V?

Answer 5

Consider the vectors v1….vm in R^n. The set of all linear combinations c1v1+….+cmvm is called their span:
span(v1,….,vm) = {c1v1+…+cmvm:c1…cm in R}

Question 6

What is the basis of the image of a given matrix A?

Answer 6

To construct a basis of the image of a matrix A, list all the column vectors of A, and omit the redundant vectors from the list.

Question 7

Demonstrate the relationship between linearly independent vectors of A and the kernel of A.

Answer 7

Ax = 0
The column vectors of A are linearly independent if (and only if) ker(A) = {0} or, equivalently, if rank(A)=m. This condition implies that m<=n

Question 8

What is a dimension?

Answer 8

Consider a subspace V of R^n. The number of vectors in a basis of V is called the dimension of V, denoted by dim(V)

Question 9

Consider a subspace V of R^n with dim(V)=m. Independent vectors and spanning vectors in a subspace of R^n. What are their properties?

Answer 9

a. We can find at most m linearly independent vectors in V
b. We need at least m vectors to span V
c. If m vectors in V are linearly independent, then they form a basis of V
d. If m vectors in V span V, then they form a basis of V.

Question 10

What is the dimension of the image equivalent to?

Answer 10

dim(imA) = rank(A) or leading 1’s

Question 11

State the rank-nullity theorem

Answer 11

For any n x n matrix A, the equation:
dim(kerA) + dim(imA) = m
holds. It can also be written as:
(nullity of A) + (rank of A) = m

Question 12

What does the dimension of the kernel say about a given matrix A?

Answer 12

dim(kerA) = number of free variables

Question 13

If A is an invertible matrix, what can you say about the rref, rank, image, and kernel of A?

Answer 13

rref(A) = I
rank(A) = n
im(A) = R^n
ker(A) = {0}