Help with simple math matrix, calculus

Question

Hi I need some help with simple math questions:)

Find a basis for the column space K, the row space R and the zero space N (the solution set of A ~ x = ~ 0) for the matrix

given the matrix

1 4 5 2

A= 2 1 3 0

-1 3 2 2.

a) Find the vector ~ u that has the K coordinates

vectormatrix [u]K= -1

-4

and the vector ~ v which has the R coordinates

vectormatrix [u]R= -2

2

b) Find the N coordinates of the vectors

-20 1

-2 -18

vector W= 14 vector T= -15

-21 4

How do you interpret the results for both vector w and t?

thanks!

Answer 1

Step 1: Column Space (K)

Pick the columns that aren't just combos of each other.

We keep columns 1, 2, and 4.

Step 2: Row Space (R)

Do the same with rows.

We end up with 3 key rows.

Step 3: Null Space (N)

Solve A⋅x=0.A \cdot x = 0.A⋅x=0.

It’s just all multiples of one vector: [−1,−1,1,0][-1, -1, 1, 0][−1,−1,1,0].