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Section MOMatrix Operations
From A First Course in Linear Algebra
Version 2.01
2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
In this section we will back up and start simple. First a definition of a totally
general set of matrices.
DefinitionVSM
Vector Space of m n
Matrices
The vector space {M}_{mn}
is the set of all m n
matrices with entries from the set of complex numbers.
(This definition contains Notation VSM.) △
Subsection MEASM: Matrix Equality, Addition, Scalar Multiplication
Just as we made, and used, a careful definition of equality for column vectors, so too, we have precise definitions for matrices.
DefinitionME
Matrix Equality
The m n
matrices A and
B are equal,
written A = B
provided {\left [A\right ]}_{ij} ={ \left [B\right ]}_{ij}
for all 1 ≤ i ≤ m,
1 ≤ j ≤ n.
(This definition contains Notation ME.) △
So equality of matrices translates to the equality of complex numbers, on an entry-by-entry basis. Notice that we now have yet another definition that uses the symbol = for shorthand. Whenever a theorem has a conclusion saying two matrices are equal (think about your objects), we will consider appealing to this definition as a way of formulating the top-level structure of the proof. We will now define two operations on the set {M}_{mn}. Again, we will overload a symbol (+) and a convention (juxtaposition for scalar multiplication).
DefinitionMA
Matrix Addition
Given the m n
matrices A
and B, define
the sum of A
and B as an
m n matrix,
written A + B,
according to
(This definition contains Notation MA.) △
So matrix addition takes two matrices of the same size and combines them (in a natural way!) to create a new matrix of the same size. Perhaps this is the obvious thing to do, but it doesnt relieve us from the obligation to state it carefully.
ExampleMA
Addition of two matrices in {M}_{23}
If
then
|
A+B = \left [\array{
2&−3& 4
\cr
1& 0 &−7 } \right ]+\left [\array{
6&2&−4
\cr
3&5& 2 } \right ] = \left [\array{
2 + 6&−3 + 2&4 + (−4)
\cr
1 + 3& 0 + 5 & −7 + 2 } \right ] = \left [\array{
8&−1& 0
\cr
4& 5 &−5 } \right ]
|