Finding Bases for subspaces of n-space

Question 1

Explain why Row space is invariant under row reduction.

Answer 1

If a matrix A is row equivalent to another matrix B, then Row(A) = Row(B). Therefore, if A is any matrix, we can transform it by elementary row operations to B, which is in RREF. Since Row(A) = Row(B), they will have the same basis.

Question 2

What do we have to do in the following problems?
The problems that are involved:
1. Given a spanning set {u1, · · · , um} for a subspace W :
(a) Finding a basis of W ,
(b) Find a basis of W , which is a subset of {u1, · · · , um}
2. Extending a given LI set {u1, · · · , um} in R^n
to a basis of R^n

Answer 2

We have to find bases for the row and columns spaces of a matrix.

Question 3

What is a fact about bases of row spaces?

Answer 3

The non-zero rows in any REF of A form a basis for
Row(A), so dim(Row(A)) = rank(A).
For the RREF of A, we call them the standard basis of the subspace Row(A).

Question 4

Define the row-space algorithm.

Find a basis of W

Answer 4

The row space algorithm starts by writing a matrix whose rows are the vectors in the given spanning set of a subspace W , i.e. W = Row(A).
Then, by finding non-zero rows of any REF (or the RREF) of A we can obtain a basis for the subspace W .

Question 5

Given a LI set {u1, · · · , um} in R^n, to extend it to a basis of R^n (assuming m < n) we: (3)

Answer 5

1. Write a n×n matrix A whose first m rows are u1,···,um and leave the n−m unknown rows um+1,···,un as symbols,
2. We know that if rank(A)=n, then the rows of A will be a basis of R^n. Hence, we row reduce the first m rows of A
   and identify the columns with leading 1s in them,
3. We choose from the standard basis of R^n
   to make sure there
   is a leading 1 in every column of A.

Question 6

Let A be a m × n matrix. Then a basis for Col(A) consists of…

Answer 6

Those columns of A which give rise to leading 1s in any
REF (or the RREF) of A. Hence, dim(Col(A)) = rank(A) = # of LI columns of A.
Note
In contrast to the row space, if A → R by elementary row operations, then Col(A)≠Col(R).

Question 7

Define the column-space algorithm.

Find a basis of W, which is a subset of {u1, · · · , um}

Answer 7

Suppose A = [a1 a2 · · · an] is written in block column format:
1. {a1}: keep it, if it is a non-zero vector,
2. {a1, a2}: If a2∈Span{a1}, then {a1, a2} is LD and we
discard a2. Otherwise, we keep a2.
3. {a1, a2, a3}: If a3 ∈ Span{a1, a2}, then {a1, a2, a3} is LD and
we discard a3. Otherwise, we keep a3.
We continue and throw out ai
if
rank([a1 · · · ai−1]) = rank([a1 · · · ai−1 | ai]),
or when there is no leading 1 in the i-th column of the RREF of A.

Question 8

What are three facts about row-space and column-space?

Answer 8

1. Since the rows of any m×n matrix A are the columns of A^T, we have Row(A) = Col(A^T) and Col(A) = Row(A^T).
2. dim(Row(A)) = dim(Col(A)) = dim(Row(A^T)) =
   dim(Col(A^T)) = rank(A) = rank(A^T ).
3. Let A be a m×n matrix with r1, · · · ,rm as its 1×n rows.
   Then,
   u∈ker(A) ⇔
   [ r1 ]
   [ … ]
   [ … ] u = 0
   [rm]
   ⇔
   [r1 . u]
   [ … ]
   [ … ]
   [ … ]
   [rm.u]
   =
   [0]
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