Independence of Rows and Columns in Matrices

[stunner5000pt]

a few questions
a) can a 3x4 matrix have independent columns? rows? Explain
if i were to reduce to row echelon form then i could potentially have 4 leading 1s. I m not quite sure about this.
if i were to reduce this 3 x 4 matrix into row echelon form then the number of rows is less than the number of variables. SO the answer is no.

b) if A is a 4 x3 matrix and rank A = 2, can A have independent columns? rows? Explain
ok rank A means that out of the 4 rows only 2 are non zero when A is in row echelon form. Potentially 3 leading 1s in the columns so at least 2 of the columns may be independent on each other. So independent columns are not possible.
Indepednat rows not possible.

c) Can a non square matrix has its rows indepedant and its columns independent?
im not sure about this. If A (MxN) then for m rows A has n unknowns so it is not possible to have indepdnatn rows. As for the columns i ahve no idea.

If A is m x n and B is n x m show taht AB = 0 iff $col B \subseteq null A$
suppose AB = 0
let columns of B = $C_{i}$
rows of A = $R_{i}$
for all i
then [tex]R_{i} C_{i} = 0 [/itex] if Ci = 0 for all i. Thus Ci belongs to null A<br /> Suppose $$col B \subseteq null A$$<br /> then anything times a column of B is zero. Thus AB = 0 <br /> Is this proof adequate?<br /> <br /> your input is greatly appreciated![/tex]

 

[TD]

Remember that a rank of a matrix is equal to the ranks of its transpose, this allows you to intechange rows & columns for your explanation.

A 3x4 matrix can have at most a rank 3, so what does that tell you about the maximum number of linearly indepedant rows/columns?
For the second, the rank is now given - what does this tell you again?
You can use the same argument again for a non-square m x n. Suppose m > n, then the maximal possible ranks is n.

 

[stunner5000pt]

for a 3x4 matrix\
the rank can be at most 3
that means it can have at most 3 linearly independent rows
4 linear independent columns

for hte second
for rank A = 2
then there are 2 indpendant rows
so at most only 2 indpendant columns?