Matrix of order supose 3*3 is in span of R3

[asad khan]

sir kindly tell me how to know that a matrix of order supose 3*3 is in span of R3 ?...

 

[HallsofIvy]

I'm afraid you will have to phrase the question more precisely. "Span" is defined for a set of vectors so I don't know what you mean by "in span of R3". Further, if you mean just "in R3", NO matrices are in R³. Everything in R³ is a vector with 3 components. (You can consider "1 by 3" or "3 by 1" matrices to be in R³ but you would have to drop the multiplication of matrices to do that.

 

[hholzer]

(Note: I use vector in a general sense: a vector can be a function or
a matrix, etc, as they represent elements in a vector space.).

a spanning set for the vector space M_3(R) -- the set of 3x3 matrices whose
elements are in the reals -- will be a set of vectors such that every vector
in M_3(R) can be written as a linear combination of the set.

Of course, such a set may contain more vectors than necessary but may
nonetheless span all of M_3(R). When such a set contains more vectors
than necessary, the set of such vectors is LINEARLY DEPENDENT.
You can attain a LINEARLY INDEPENDENT set of vectors by removing
those vectors in your set that can be written as a linear combination
of the other vectors.

When you attain such a LINEARLY INDEPENDENT set of vectors, you
acquire a spanning set called a MINIMAL SPANNING SET.

For M_3(R), you will need AT LEAST 9 vectors in your set(but again,
you can have more). Furthermore, such a set of AT LEAST 9 vectors
is not unique.

So how do you determine if a 3x3 matrix spans M_3(R) -- assuming this
is what you meant to ask?
Well, you can't. You'll need AT LEAST 9 matrices to do this.
Once you have 9 (or more) matrices, you need to determine if
v = (c_1)v_1 + (c_2)v_2 + ... + (c_n)v_n (n >= 9)

if you can determine that this system of equations has
a unique solution, then you can determine that the set of
AT LEAST 9 vectors spans M_3(R).