Consider the following matrix A and suppose the matrix U is row equivalent to A

ID: 2970541 • Letter: C

Question

Consider the following matrix A and suppose the matrix U is row equivalent to A.

3 2 1 4 1

0 0 -1 -2 1

-3 -5 2 2 1

0 3 -2 -4 1

U =

3 2 1 4 1

0 -3 3 6 2

0 0 -1 -2 1

0 0 0 0 4

Compute each of the following (you may not want to do the calculations

Be sure to state any pertinent facts you are using.

(1) Find the null space of A, Null(A), and compute its dimension.

(2) Find a basis for the row space of A.

(3) What is Rank(A)?

(4) Find a basis for the column space of A.

(5) Do the columns of A span R4? (why or why not?)

Write short phrases to explain your work.

a) Solving Ux=0 with x=(a b c d e)^T gives ( starting from bottom )

e=0, -c-2d=0, b=0,3a+c+4d=3a+2d=0, so c=3a and d=-3/2a

So (a,0,3a,-3/2a,0) so we can take as basis (2,0,6,-3,0)

b) since U has no zero row, A is full rank and the row space is each row of A

c) rank(A)=4 since it's full rank

d) the column space has same dimension than row space, so 4.

We see from U that the 4th column is 2x 3rd column + 2/3 first column

So the basis of the column space are the 1st,2nd,3rd and 5th column

e) yes, because same dimension and col space is a subspace of R^4, so they are equal

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